EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.
Solitons in nonlocal nonlinear media: Exact solutions
DEFF Research Database (Denmark)
Krolikowski, Wieslaw; Bang, Ole
2001-01-01
We investigate the propagation of one-dimensional bright and dark spatial solitons in a nonlocal Kerr-like media, in which the nonlocality is of general form. We find an exact analytical solution to the nonlinear propagation equation in the case of weak nonlocality. We study the properties...
Exact solutions for nonlinear foam drainage equation
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2016-09-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G) -expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Exact solutions for nonlinear foam drainage equation
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2017-02-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G)-expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
A Family of Exact Solutions for the Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLSequation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these sta-tionary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.
The exact solutions for a nonisospectral nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Ning Tongke [Finance College, Shanghai Normal University, Shanghai 200234 (China)], E-mail: tkning@shnu.edu.cn; Zhang Weiguo; Jia Gao [Science College, University of Shanghai for Science and Technology, Shanghai 200093 (China)
2009-10-30
In this paper, lax pair for the nonisospectral nonlinear Schroedinger hierarchy is given, the time dependence of nonisospectral scattering data is derived and exact solutions for the nonisospectral nonlinear Schroedinger hierarchy are obtained through the inverse scattering transform.
Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2013-01-01
Full Text Available We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.
Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation[
Institute of Scientific and Technical Information of China (English)
HUANGDing-Jiang; ZHANGHong-Qing
2004-01-01
By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.
Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation
Institute of Scientific and Technical Information of China (English)
HUANG Ding-Jiang; ZHANG Hong-Qing
2004-01-01
By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.
Exact Solution of a Generalized Nonlinear Schrodinger Equation Dimer
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Maniadis, P.; Tsironis, G.P.
1998-01-01
We present exact solutions for a nonlinear dimer system defined throught a discrete nonlinear Schrodinger equation that contains also an integrable Ablowitz-Ladik term. The solutions are obtained throught a transformation that maps the dimer into a double Sine-Gordon like ordinary nonlinear...... differential equation....
Exact solitary wave solutions of nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grbner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.
Exact periodic wave solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt); Elgarayhi, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: elgarayhi@yahoo.com; Elhanbaly, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)
2006-08-15
The periodic wave solutions for some nonlinear partial differential equations, including generalized Klein-Gordon equation, Kadomtsev-Petviashvili (KP) equation and Boussinesq equations, are obtained by using the solutions of Jacobi elliptic equation. Under limit conditions, exact solitary wave solutions, shock wave solutions and triangular periodic wave solutions have been recovered.
Exact solutions for the cubic-quintic nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Zhu Jiamin [Department of Physics, Zhejiang Lishui University, Lishui 323000 (China)]. E-mail: zjm64@163.com; Ma Zhengyi [Department of Physics, Zhejiang Lishui University, Lishui 323000 (China); Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072 (China)
2007-08-15
In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.
Exact travelling wave solutions for some important nonlinear physical models
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2013-05-01
The two-dimensional nonlinear physical models and coupled nonlinear systems such as Maccari equations, Higgs equations and Schrödinger–KdV equations have been widely applied in many branches of physics. So, finding exact travelling wave solutions of such equations are very helpful in the theories and numerical studies. In this paper, the Kudryashov method is used to seek exact travelling wave solutions of such physical models. Further, three-dimensional plots of some of the solutions are also given to visualize the dynamics of the equations. The results reveal that the method is a very effective and powerful tool for solving nonlinear partial differential equations arising in mathematical physics.
Exact solutions for nonlinear partial fractional differential equations
Institute of Scientific and Technical Information of China (English)
Khaled A.Gepreel; Saleh Omran
2012-01-01
In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved (G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
A procedure to construct exact solutions of nonlinear evolution equations
Indian Academy of Sciences (India)
Adem Cengiz Çevikel; Ahmet Bekir; Mutlu Akar; Sait San
2012-09-01
In this paper, we implemented the functional variable method for the exact solutions of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW), the modified Benjamin-Bona-Mohany (mBBM) and the modified kdV-Kadomtsev-Petviashvili (kdV-KP) equation. By using this scheme, we found some exact solutions of the above-mentioned equation. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. The functional variable method presents a wider-applicability for handling nonlinear wave equations.
REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
YeCaier; PanZuliang
2003-01-01
Nonlinear partial differetial equation(NLPDE)is converted into ordinary differential equation(ODE)via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
Sirendaoreji
2004-01-01
Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.
Exact periodic solution in coupled nonlinear Schrodinger equations
Institute of Scientific and Technical Information of China (English)
Li Qi-Liang; Chen Jun-Lang; Sun Li-Li; Yu Shu-Yi; Qian Sheng
2007-01-01
The coupled nonlinear Schrodinger equations (CNLSEs) of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.
Exact solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Peng, Yan-Ze
2003-08-11
Exact solutions to some nonlinear partial differential equations, including (2+1)-dimensional breaking soliton equation, sine-Gordon equation and double sine-Gordon equation, are studied by means of the mapping method proposed by the author recently. Many new results are presented. A simple review of the method is finally given.
Exact solutions to a nonlinear dispersive model with variable coefficients
Energy Technology Data Exchange (ETDEWEB)
Yin Jun [Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074 (China); Lai Shaoyong [Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074 (China)], E-mail: laishaoy@swufe.edu.cn; Qing Yin [Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074 (China)
2009-05-15
A mathematical technique based on an auxiliary differential equation and the symbolic computation system Maple is employed to investigate a prototypical and nonlinear K(n, n) equation with variable coefficients. The exact solutions to the equation are constructed analytically under various circumstances. It is shown that the variable coefficients and the exponent appearing in the equation determine the quantitative change in the physical structures of the solutions.
New Exact Solutions for New Model Nonlinear Partial Differential Equation
Directory of Open Access Journals (Sweden)
A. Maher
2013-01-01
Full Text Available In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.
Exact travelling wave solutions of nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt)]. E-mail: asoliman_99@yahoo.com; Abdou, M.A. [Theoretical Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-04-15
An extended Fan-sub equation method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The key idea of this method is to take full advantage of the general elliptic equation, involving five parameters, which has more new solutions and whose degeneracies can lead to special sub equation involving three parameters. As an illustration of the extended Fan method, more new solutions are obtained for three models namely, generalized KdV, Drinfeld-Sokolov system and RLW equation.
Exact solutions of certain nonlinear chemotaxis diffusion reaction equations
Indian Academy of Sciences (India)
MISHRA AJAY; KAUSHAL R S; PRASAD AWADHESH
2016-05-01
Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in theconstructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.
DEFF Research Database (Denmark)
Khare, A.; Rasmussen, K. O.; Samuelsen, Mogens Rugholm
2010-01-01
We show that the two-dimensional, nonlinear Schrodinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the e......We show that the two-dimensional, nonlinear Schrodinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show...
Khare, Avinash; Samuelsen, Mogens R; Saxena, Avadh; 10.1088/1751-8113/43/37/375209
2010-01-01
We show that the two-dimensional, nonlinear Schr\\"odinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero.
Time-dependent exact solutions of the nonlinear Kompaneets equation
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, N H, E-mail: nib@bth.s [Department of Mathematics and Science, Blekinge Institute of Technology, 371 79 Karlskrona (Sweden)
2010-12-17
Time-dependent exact solutions of the Kompaneets photon diffusion equation are obtained for several approximations of this equation. One of the approximations describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact solutions as group invariant solutions. (fast track communication)
Exact solutions for some nonlinear systems of partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)], E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)], E-mail: aramady@yahoo.com
2009-04-30
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear systems of partial differential equations (PDEs) is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDEs) are obtained. Graphs of the solutions are displayed.
New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
YANG Qin; DAI Chao-Qing; ZHANG Jie-Fang
2005-01-01
Some new exact travelling wave and period solutions of discrete nonlinear Schrodinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differentialdifferent models.
Analytical exact solution of the non-linear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da [Universidade de Brasilia (UnB), DF (Brazil). Inst. de Fisica. Grupo de Fisica e Matematica
2011-07-01
Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)
New explicit exact solutions to a nonlinear dispersive-dissipative equation
Institute of Scientific and Technical Information of China (English)
Naranmandula; Wang Ke-Xie
2004-01-01
Using the first-integral method, we obtain a series of new explicit exact solutions such as exponential function solutions, triangular function solutions, singular solitary wave solution and kink solitary wave solution of a nonlinear dispersive-dissipative equation, which describes weak nonlinear ion-acoustic waves in plasma consisting of cold ions and warm electrons.
Exact solutions of some nonlinear partial differential equations using functional variable method
Indian Academy of Sciences (India)
A Nazarzadeh; M Eslami; M Mirzazadeh
2013-08-01
The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm Kadomtsev–Petviashvili equation and the higher-order nonlinear Schrödinger equation. By using this useful method, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. It is shown that the proposed method is effective and general.
EXACT SOLITARY WAVE SOLUTIONS OF THETWO NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
ZhuYanjuan; ZhangChunhua
2005-01-01
The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dimensional nonlinear evolution equations.
Exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations
Energy Technology Data Exchange (ETDEWEB)
Sirendaoreji [College of Mathematical Science, Inner Mongolia Normal University, Huhhot 010022, Inner Mongolia (China)]. E-mail: siren@imnu.edu.cn
2007-04-09
A variable separated equation and its solutions are used to construct the exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations. The solutions previously obtained by the tanh and sech method are recovered. New and more exact travelling wave solutions including solitons, kink and anti-kink, bell and anti-bell solitary wave solutions, periodic solutions, singular solutions and exponential solutions are found.
Nonlinear differential equations with exact solutions expressed via the Weierstrass function
Kudryashov, NA
2004-01-01
A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear di
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.
New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup Equation
Directory of Open Access Journals (Sweden)
Zhenshu Wen
2014-01-01
Full Text Available We study the nonlinear wave solutions for the Broer-Kaup equation. Many exact explicit expressions of the nonlinear wave solutions for the equation are obtained by exploiting the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which are new. Some previous results are extended.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H
2015-07-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.
Exact discrete soliton solutions of quintic discrete nonlinear Schr(o)dinger equation
Institute of Scientific and Technical Information of China (English)
Li Hua-Mei; Wu Feng-Min
2005-01-01
By using the extended hyperbolic function approach, we have studied a quintic discrete nonlinear Schrodinger equation and obtained new exact localized solutions, including the discrete bright soliton solution, dark soliton solution,alternating phase bright soliton solution and alternating phase dark soliton solution, if a special constraint is imposed on the coefficients of the equation.
New Exact Solutions for a Class of Nonlinear Coupled Differential Equations
Institute of Scientific and Technical Information of China (English)
ZHAO Hong; GUO Jun; BAI Cheng-Lin; HAN Ji-Guang
2005-01-01
More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance method.
Exact bright and dark spatial soliton solutions in saturable nonlinear media
Energy Technology Data Exchange (ETDEWEB)
Calvo, Gabriel F. [Departamento de Matematicas, E.T.S. de Ingenieros Industriales, Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), E.T.S.I. Industriales, Avda. Camilo Jose Cela, 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain); Belmonte-Beitia, Juan [Departamento de Matematicas, E.T.S. de Ingenieros Industriales, Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), E.T.S.I. Industriales, Avda. Camilo Jose Cela, 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: Juan.Belmonte@uclm.es; Perez-Garcia, Victor M. [Departamento de Matematicas, E.T.S. de Ingenieros Industriales, Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), E.T.S.I. Industriales, Avda. Camilo Jose Cela, 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)
2009-08-30
We present exact analytical bright and dark (black and grey) solitary wave solutions of a nonlinear Schroedinger-type equation describing the propagation of spatial beams in media exhibiting a saturable nonlinearity (such as centrosymmetric photorefractive materials). A qualitative study of the stationary equation is carried out together with a discussion of the stability of the solutions.
STUDY ON EXACT ANALYTICAL SOLUTIONS FOR TWO SYSTEMS OF NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
闫振亚; 张鸿庆
2001-01-01
The homogeneous balance method was improved and applied to two systems of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.
Institute of Scientific and Technical Information of China (English)
ZHANGJin-Liang; WANGMing-Liang
2004-01-01
The complex tanh-function expansion method was presented recently, and it can be applied to derive exact solutions to the Schroedinger-type nonlinear evolution equations directly without transformation. In this paper,the complex tanh-function expansion method is applied to derive the exact solutions to the general coupled nonlinear evolution equations. Zakharov system and a long-short-wave interaction system are considered as examples, and the new applications of the complex tanh-function expansion method are shown.
Institute of Scientific and Technical Information of China (English)
ZHANG Jin-Liang; WANG Ming-Liang
2004-01-01
The complex tanh-function expansion method was presented recently, and it can be applied to derive exact solutions to the Schrodinger-type nonlinear evolution equations directly without transformation. In this paper,the complex tanh-function expansion method is applied to derive the exact solutions to the general coupled nonlinear evolution equations. Zakharov system and a long-short-wave interaction system are considered as examples, and the new applications of the complex tanh-function expansion method are shown.
The exact solutions to (2+1)-dimensional nonlinear Schrǒdinger equation
Institute of Scientific and Technical Information of China (English)
ZHANG Jin-liang; WANG Ming-liang; FANG Zong-de
2004-01-01
By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrǒdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.
Exact Solutions for a Higher-Order Nonlinear Schr(o)dinger Equation in Atmospheric Dynamics
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
By giving prior assumptions on the form of the solutions, we succeed to find several exact solutions for a higher-order nonlinear Schrodinger equation derived from one important model in the study of atmospheric and ocean dynamical systems. Our analytical solutions include bright and dark solitary waves, and periodical solutions, which can be used to explain atmospheric phenomena.
Lienard Equation and Exact Solutions for Some Soliton-Producing Nonlinear Equations
Institute of Scientific and Technical Information of China (English)
ZHANG Wei-Guo; CHANG Qian-Shun; ZHANG Qi-Ren
2004-01-01
In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.
Tian, Qing; Wu, Lei; Zhang, Jie-Fang; Malomed, Boris A; Mihalache, D; Liu, W M
2011-01-01
We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.
Directory of Open Access Journals (Sweden)
Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
Regarding on the exact solutions for the nonlinear fractional differential equations
Directory of Open Access Journals (Sweden)
Kaplan Melike
2016-01-01
Full Text Available In this work, we have considered the modified simple equation (MSE method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW and the modified equal width (mEW equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.
Exact Solutions to Extended Nonlinear Schr(o)dinger Equation in Monomode Optical Fiber
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin; ZHAO Hong; Wang Wei-Tao
2006-01-01
By using the generally projective Riccati equation method, more new exact travelling wave solutions to extended nonlinear Schr(o)dinger equation (NLSE), which describes the femtosecond pulse propagation in monomode optical fiber, are found, which include bright soliton solution, dark soliton solution, new solitary waves, periodic solutions, and rational solutions. The finding of abundant solution structures for extended NLSE helps to study the movement rule of femtosecond pulse propagation in monomode optical fiber.
Indian Academy of Sciences (India)
M Mirzazadeh; M Eslami
2013-12-01
Studying compactons, solitons, solitary patterns and periodic solutions is important in nonlinear phenomena. In this paper we study nonlinear variants of the Kadomtsev–Petviashvili (KP) and the Korteweg–de Vries (KdV) equations with positive and negative exponents. The functional variable method is used to establish compactons, solitons, solitary patterns and periodic solutions for these variants. This method is a powerful tool for searching exact travelling solutions in closed form.
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities
Beléndez,Augusto; Beléndez Vázquez, Tarsicio; Martínez Guardiola, Francisco Javier; Pascual Villalobos, Carolina; Álvarez López, Mariela Lázara; Arribas Garde, Enrique
2016-01-01
The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, unforced cubic–quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobi elliptic functions. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the per...
Indian Academy of Sciences (India)
Wenjun Liu; Kewang Chen
2013-09-01
In this paper, we implemented the functional variable method and the modified Riemann–Liouville derivative for the exact solitary wave solutions and periodic wave solutions of the time-fractional Klein–Gordon equation, and the time-fractional Hirota–Satsuma coupled KdV system. This method is extremely simple but effective for handling nonlinear time-fractional differential equations.
Directory of Open Access Journals (Sweden)
Yusuf Pandir
2012-01-01
Full Text Available We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.
Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations
Energy Technology Data Exchange (ETDEWEB)
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt); Department of Mathematics, Teacher' s College, Bisha, P.O. Box 551 (Saudi Arabia)], E-mail: asoliman_99@yahoo.com
2007-08-27
By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, Juan [Departamento de Matematicas, E.T.S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), Avda. Camilo Jose Cela 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: juan.belmonte@uclm.es; Calvo, Gabriel F. [Departamento de Matematicas, E.T.S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), Avda. Camilo Jose Cela 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: gabriel.fernandez@uclm.es
2009-01-19
In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schroedinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions.
The exact solutions of nonlinear problems by Homotopy Analysis Method (HAM
Directory of Open Access Journals (Sweden)
Hafiz Abdul Wahab
2016-06-01
Full Text Available The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy Analysis method gives the same series solution as in Adomian Decomposition Method as well as homotopy Perturbation Method (Wahab et al, 2015 and we get the exact solution using the initial guess in Homotopy Analysis Method using the results obtained by Adomian Decomposition Method.
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
A new procedure is proposed to construct strongly nonlinear systems of multiple degrees of freedom subjected to parametric and/or external Gaussian white noises, whose exact stationary solutions are independent of energy. Firstly, the equivalent Fokker-Planck-Kolmogorov (FPK) equations are derived by using exterior differentiation. The main difference between the equivalent FPK equation and the original FPK equation lies in the additional arbitrary antisymmetric diffusion matrix. Then the exact stationary solutions and the structures of the original systems can be obtained by using the coefficients of antisymmetric diffusion matrix. The obtained exact stationary solutions, which are generally independent of energy, are for the most general class of strongly nonlinear stochastic systems multiple degrees of freedom (MDOF) so far, and some classes of the known ones dependent on energy belong to the special cases of them.
Institute of Scientific and Technical Information of China (English)
HUANG ZhiLong; JIN XiaoLing
2009-01-01
A new procedure is proposed to construct strongly nonlinear systems of multiple degrees of freedom subjected to parametric and/or external Gaussian white noises,whose exact stationary solutions are independent of energy.Firstly,the equivalent Fokker-Planck-Kolmogorov(FPK)equations are derived by using exterior differentiation.The main difference between the equivalent FPK equation and the original FPK equation lies in the additional arbitrary antisymmetric diffusion matrix.Then the exact stationary solutions and the structures of the original systems can be obtained by using the coefficients of antisymmetric diffusion matrix.The obtained exact stationary solutions,which are generally independent of energy,are for the most general class of strongly nonlinear stochastic systems multiple degrees of freedom(MDOF)so far,and some classes of the known ones dependent on energy belong to the special cases of them.
A procedure to construct exact solutions of nonlinear fractional differential equations.
Güner, Özkan; Cevikel, Adem C
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.
Directory of Open Access Journals (Sweden)
Md. Nur Alam
2016-06-01
Full Text Available In this article, we apply the exp(-Φ(ξ-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.
Institute of Scientific and Technical Information of China (English)
李志斌; 陈天华
2002-01-01
An algorithm for constructing exact solitary wave solutions and singular solutions for a class of nonlinear dissipative-dispersive system is presented. With the aid of symbolic manipulation system Maple, some explicit solutions are obtained for the system in physically interesting but non-integrable cases.
Exact Solutions of Some (1+1)-Dimensional Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
By means of the variable separation method, new exact solutions of some (1+1)-dimensional nonlinear evolution equations are obtained. Abundant localized excitations can be found by selecting corresponding arbitrary functions appropriately. Namely, the new soliton-like localized excitations and instanton-like localized excitations are presented.
Directory of Open Access Journals (Sweden)
Yusuf Pandir
2013-01-01
Full Text Available We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE, we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.
Coupled Nonlinear Schr(o)dinger Equation: Symmetries and Exact Solutions
Institute of Scientific and Technical Information of China (English)
LIU Ping; LOU Sen-Yue
2009-01-01
The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schr(o)dinger (CNLS) equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.
Indian Academy of Sciences (India)
R S Kaushal; Ranjit Kumar; Awadhesh Prasad
2006-08-01
Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.
Exact solutions of SO(3) non-linear sigma model in a conic space background
Bezerra, V B; Romero, C
2005-01-01
We consider a nonlinear sigma model coupled to the metric of a conic space. We obtain restrictions for a nonlinear sigma model to be a source of the conic space. We then study nonlinear sigma model in the conic space background. We find coordinate transformations which reduce the chiral fields equations in the conic space background to field equations in Minkowski spacetime. This enables us to apply the same methods for obtaining exact solutions in Minkowski spacetime to the case of a conic spacetime. In the case the solutions depend on two spatial coordinates we employ Ivanov's geometrical ansatz. We give a general analysis and also present classes of solutions in which there is dependence on three and four coordinates. We discuss with special attention the intermediate instanton and meron solutions and their analogous in the conic space. We find differences in the total actions and topological charges of these solutions and discuss the role of the deficit angle.
Exact travelling solutions for some nonlinear physical models by (′/)-expansion method
Indian Academy of Sciences (India)
B Salim Bahrami; H Abdollahzadeh; I M Berijani; D D Ganji; M Abdollahzadeh
2011-08-01
In this paper, we establish exact solutions for some special nonlinear partial differential equations. The (′/)-expansion method is used to construct travelling wave solutions of the twodimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrödinger–KdV equations, which appear in many ﬁelds such as, solid-state physics, nonlinear optics, ﬂuid dynamics, ﬂuid ﬂow, quantum ﬁeld theory, electromagnetic waves and so on. In this method we take the advantage of general solutions of second-order linear ordinary differential equation (LODE) to solve many nonlinear evolution equations effectively. The (′/)-expansion method is direct, concise and elementary and can be used with a wider applicability for handling many nonlinear wave equations.
Construction and exact solution of a nonlinear quantum field model in quasi-higher dimension
Energy Technology Data Exchange (ETDEWEB)
Kundu, Anjan, E-mail: anjan.kundu@saha.ac.in
2015-10-15
Nonperturbative exact solutions are allowed for quantum integrable models in one space-dimension. Going beyond this class we propose an alternative Lax matrix approach, exploiting the hidden multi-space–time concept in integrable systems and construct a novel nonlinear Schrödinger quantum field model in quasi-two dimensions. An intriguing field commutator is discovered, confirming the integrability of the model and yielding its exact Bethe ansatz solution with rich scattering and bound-state properties. The universality of the scheme is expected to cover diverse models, opening up a new direction in the field.
Exact Solutions to Maccari's System
Institute of Scientific and Technical Information of China (English)
PAN Jun-Ting; GONG Lun-Xun
2007-01-01
Based on the generalized Riccati relation, an algebraic method to construct a series of exact solutions to nonlinear evolution equations is proposed. Being concise and straightforward, the method is applied to Maccari's system, and some exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.
Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua
2016-12-01
Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205
Exact Solutions of Nonlinear Dynamics Equation in a New Double-Chain Model of DNA
Institute of Scientific and Technical Information of China (English)
QIAN Xian-Min; LOU Sen-Yue
2003-01-01
The exact solutions of the general nonlinear dynamic system in a new double-chain model of DNA are studiedkink shape excitations can be found in both the Conte's truncation expansion and the Pickering's truncation expansion.Three types of new localized excitations, the asymmetric kink-kink excitations, the soliton-kink excitation, and thekink-soliton excitations, are found by using the Pickering's nonstandard truncation expansion.
Exact solutions to a class of nonlinear Schrödinger-type equations
Indian Academy of Sciences (India)
Jin-Liang Zhang; Ming-Liang Wing
2006-12-01
A class of nonlinear Schrödinger-type equations, including the Rangwala–Rao equation, the Gerdjikov–Ivanov equation, the Chen–Lee–Lin equation and the Ablowitz–Ramani–Segur equation are investigated, and the exact solutions are derived with the aid of the homogeneous balance principle, and a set of subsidiary higher order ordinary differential equations (sub-ODEs for short).
Multi-Order Exact Solutions for a generalized shallow water wave equation and other nonlinear PDEs
Bagchi, Bijan; Ganguly, Asish
2011-01-01
We seek multi-order exact solutions of a generalized shallow water wave equation along with those corresponding to a class of nonlinear systems described by the KdV, modified KdV, Boussinesq, Klein-Gordon and modified Benjamin-Bona-Mahony equation. We employ a modified version of a generalized Lame equation and subject it to a perturbative treatment identifying the solutions order by order in terms of Jacobi elliptic functions. Our solutions are new and hold the key feature that they are expressible in terms of an auxiliary function f in a generic way. For appropriate choices of f we recover the previous results reported in the literature.
EXACT SOLUTIONS FOR NONLINEAR TRANSIENT FLOW MODEL INCLUDING A QUADRATIC GRADIENT TERM
Institute of Scientific and Technical Information of China (English)
曹绪龙; 同登科; 王瑞和
2004-01-01
The models of the nonlinear radial flow for the infinite and finite reservoirs including a quadratic gradient term were presented. The exact solution was given in real space for flow equation including quadratic gradiet term for both constant-rate and constant pressure production cases in an infinite system by using generalized Weber transform. Analytical solutions for flow equation including quadratic gradient term were also obtained by using the Hankel transform for a finite circular reservoir case. Both closed and constant pressure outer boundary conditions are considered. Moreover, both constant rate and constant pressure inner boundary conditions are considered. The difference between the nonlinear pressure solution and linear pressure solution is analyzed. The difference may be reached about 8% in the long time. The effect of the quadratic gradient term in the large time well test is considered.
Nonlinear wave structures as exact solutions of Vlasov-Maxwell equations.
Dasgupta, B.; Tsurutani, B. T.; Janaki, M. S.; Sharma, A. S.
2001-12-01
Many recent observations by POLAR and Geotail spacecraft of the low-latitudes magnetopause boundary layer (LLBL) and the polar cap boundary layer (PCBL) have detected nonlinear wave structures [Tsurutani et al, Geophys. Res. Lett., 25, 4117, 1998]. These nonlinear waves have electromagnetic signatures that are identified with Alfven and Whistler modes. Also solitary waves with mono- and bi-polar features were observed. In general such electromagnetic structures are described by the full Vlasov-Maxwell equations for waves propagating at an angle to the ambient magnetic field, but it has been a diffficult task obtaining the solutions because of the inherent nonlinearity. We have obtained an exact nonlinear solution of the full Vlasov-Maxwell equations in the presence of an electromagnetic wave propagating at an arbitrary direction with an ambient magnetic field. This is accomplished by finding the constants of motion of the charged particles in the electromagnetic field of the wave and then constructing a realistic distribution function as a function of these constants of motion. The corresponding trapping conditions for such waves are obtained, yielding the self-consistent description for the particles in the presence of the nonlinear waves. The interpretation of the observed nonlinear structures in terms of these general solutions will be presented.
Institute of Scientific and Technical Information of China (English)
Li Hua-Mei
2005-01-01
By using the mapping method and an appropriate transformation, we find new exact solutions of nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions , triangular function solutions, bright and dark solitons, and soliton-like solutions.
Institute of Scientific and Technical Information of China (English)
JIANG Zhi-ping
2012-01-01
With the help of the variable-coefficient generalized projected Ricatti equation expansion method,we present exact solutions for the generalized (2+1)-dimensional nonlinear Schr(o)dinger equation with variable coefficients.These solutions include solitary wave solutions,soliton-like solutions and trigonometric function solutions.Among these solutions,some are found for the first time.
Energy Technology Data Exchange (ETDEWEB)
Mvogo, Alain, E-mail: mvogal_2009@yahoo.fr [Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I (Cameroon); Ben-Bolie, G.H., E-mail: gbenbolie@yahoo.fr [Laboratory of Nuclear Physics, Department of Physics, Faculty of Science, University of Yaounde I (Cameroon); Centre d' Excellence Africain en Technologies de l' Information et de la Communication, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon); Kofané, T.C., E-mail: tckofane@yahoo.com [Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I (Cameroon); Centre d' Excellence Africain en Technologies de l' Information et de la Communication, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon); The Abdus Salam International Center for Theoretical Physics, P.O. Box 586, Strada Costiera 11, I-34014 Trieste (Italy)
2014-07-04
An improved quantum model for exciton–phonon dynamics in an α-helix is investigated taking into account the interspine coupling and the influence of power-law long-range exciton–exciton interactions. Having constructed the model Hamiltonian, we derive the lattice equations and employ the Fourier transforms to go in continuum space showing that the long-range interactions (LRI) lead to a nonlocal integral term in the equations of motion. Indeed, the non-locality originating from the LRI results in the dynamic equations with space derivatives of fractional order. New theoretical frameworks are derived, such that: fractional generalization of coupled Zakharov equations, coupled nonlinear fractional Schrödinger equations, coupled fractional Ginzburg–Landau equations, coupled Hilbert–Zakharov equations, coupled nonlinear Hilbert–Ginzburg–Landau equations, coupled nonlinear Schrödinger equations and coupled nonlinear Hilbert–Schrödinger equations. Through the F-expansion method, we derive a set of exact Jacobian solutions of coupled nonlinear Schrödinger equations. These solutions include Jacobian periodic solutions as well as bright and dark soliton which are important in the process of energy transport in the molecule. We also discuss of the impact of LRI on the energy transport in the molecule.
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE, (2 + 1-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.
非线性系统的精确解%Constructing Exact Solutions for Two Nonlinear Systems
Institute of Scientific and Technical Information of China (English)
赵雪芹; 智红燕; 张鸿庆
2008-01-01
Based on the computerized symbolic,a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (PDES) in a unified way.The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions.At the same time,we present a more general transformation,which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations (NLEEs).More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.
Institute of Scientific and Technical Information of China (English)
ZHU Jia-Min; LIU Yu-Lu
2009-01-01
By constructing appropriate transformations and an extended elliptic sub-equation approach, we find some exact solutions of variable coefficient cubic-qulntic nonlinear Schrodinger equation with an external potential, which include bell and kink profile solitary wave solutions, singular solutions, triangular periodic wave solutions and so on.
Institute of Scientific and Technical Information of China (English)
LI Hua-Mei
2003-01-01
In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.
Directory of Open Access Journals (Sweden)
Weiguo Rui
2014-01-01
Full Text Available By using the integral bifurcation method together with factoring technique, we study a water wave model, a high-order nonlinear wave equation of KdV type under some newly solvable conditions. Based on our previous research works, some exact traveling wave solutions such as broken-soliton solutions, periodic wave solutions of blow-up type, smooth solitary wave solutions, and nonsmooth peakon solutions within more extensive parameter ranges are obtained. In particular, a series of smooth solitary wave solutions and nonsmooth peakon solutions are obtained. In order to show the properties of these exact solutions visually, we plot the graphs of some representative traveling wave solutions.
Directory of Open Access Journals (Sweden)
Yong Huang
2012-01-01
Full Text Available The Bäcklund transformations and abundant exact explicit solutions for a class of nonlinear wave equation are obtained by the extended homogeneous balance method. These solutions include the solitary wave solution of rational function, the solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. In addition to rederiving some known solutions, some entirely new exact solutions are also established. Explicit and exact particular solutions of many well-known nonlinear evolution equations which are of important physical significance, such as Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system, are obtained as special cases.
Malacarne, L C; Mendes, R S; Pedron, I T; Lenzi, E K
2001-03-01
The nonlinear diffusion equation partial delta rho/delta t=D Delta rho(nu) is analyzed here, where Delta[triple bond](1/r(d-1))(delta/delta r)r(d-1-theta) delta/delta r, and d, theta, and nu are real parameters. This equation unifies the anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [ theta>(1-nu)d], "normal" diffusion [theta=(1-nu)d] and superdiffusion [theta<(1-nu)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.
Institute of Scientific and Technical Information of China (English)
柳银萍; 李志斌
2003-01-01
Based on a 0 of elliptic equation, a new algebraic method to construct a series of exact solutions for nonlinear evolution equations is proposed, meanwhile, its complete implementation TRWS in Maple is presented. The TRWS can output a series of travelling wave solutions entirely automatically, which include polynomial solutions, exponential function solutions, triangular function solutions, hyperbolic function solutions, rational function solutions, Jacobi elliptic function solutions, and Weierstrass elliptic function solutions. The effectiveness of the package is illustrated by applying it to a variety of equations. Not only are previously known solutions recovered but also new solutions and more general form of solutions are obtained.
Exact solutions of optical pulse propagation in nonlinear meta-materials
Nanda, Lipsa
2017-01-01
An analytical and simulation based method has been used to exactly solve the nonlinear wave propagation in bulk media exhibiting frequency dependent dielectric susceptibility and magnetic permeability. The method has been further extended to investigate the intensity distribution in a nonlinear meta-material with negative refractive index where both ɛ and μ are dispersive and negative in nature.
Energy Technology Data Exchange (ETDEWEB)
Yao Yuqin [College of Sciences, Shanghai University, Shanghai 200436 (China)] e-mail: yyqinw@126.com
2005-11-01
In this paper, based on the well-known Sine-Poisson equation, a new Sine-Poisson equation expansion method with constant coefficients or variable coefficients is presented, which can be used to construct more new exact solutions of nonlinear evolution equations in mathematical physics. The KdV-mKdV equation and the typical breaking soliton equation are chosen to illustrate our method such that many types of new exact solutions are obtained, which include exponential solutions, kink-shaped solutions, singular solutions and soliton-like solutions.
Kong, Chao; Hai, Kuo; Tan, Jintao; Chen, Hao; Hai, Wenhua
2016-03-01
Nonlinear Kronig-Penney model has been frequently employed to study transmission problem of electron wave in a doped semiconductor superlattice or in a nonlinear electrified chain. Here from an integral equation we derive a novel exact solution of the problem, which contains a simple nonlinear map connecting transmission coefficient with system parameters. Consequently, we propose a scheme to manipulate electronic distribution and transmission by adjusting the system parameters. A new quantum coherence effect is evidenced by the strict expression of transmission coefficient, which results in the aperiodic electronic distributions and different transmission coefficients including the approximate zero transmission and total transmission, and the multiple transmissions. The method based on the concise exact solution can be applied directly to some nonlinear cold atomic systems and a lot of linear Kronig-Penney systems, and also can be extended to investigate electronic transport in different discrete nonlinear systems.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
Using trial equation method, abundant exact envelope traveling wave solutions of high-order dispersive cubic-quintic nonlinear Schrodinger equation, which include envelope soliton solutions, triangular function envelope solutions, and Jacobian elliptic function envelope solutions, are obtained. To our knowledge, all of these results are new.In particular, our proposed method is very simple and can be applied to a lot of similar equations.
Directory of Open Access Journals (Sweden)
Yinghui He
2014-01-01
Full Text Available The construction of exact solution for higher-dimensional nonlinear equation plays an important role in knowing some facts that are not simply understood through common observations. In our work, (4+1-dimensional nonlinear Fokas equation, which is an important physical model, is discussed by using the extended F-expansion method and its variant. And some new exact solutions expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.
Exact solutions of optical wave propagation in nonlinear negative refractive medium
Nanda, Lipsa
2016-04-01
An analytical and simulation based method has been used to exactly solve the nonlinear Schrödinger's equation (NLSE) and study the solitonic forms in a medium which exhibits frequency dependent dielectric permittivity (ɛ) and magnetic permeability (μ). The model has been extended to describe the propagation of a wave in a nonlinear negative refractive medium (NRM) which is dispersive and negative in nature.
Snijkers, F.
2016-03-31
We report upon the characterization of the steady-state shear stresses and first normal stress differences as a function of shear rate using mechanical rheometry (both with a standard cone and plate and with a cone partitioned plate) and optical rheometry (with a flow-birefringence setup) of an entangled solution of asymmetric exact combs. The combs are polybutadienes (1,4-addition) consisting of an H-skeleton with an additional off-center branch on the backbone. We chose to investigate a solution in order to obtain reliable nonlinear shear data in overlapping dynamic regions with the two different techniques. The transient measurements obtained by cone partitioned plate indicated the appearance of overshoots in both the shear stress and the first normal stress difference during start-up shear flow. Interestingly, the overshoots in the start-up normal stress difference started to occur only at rates above the inverse stretch time of the backbone, when the stretch time of the backbone was estimated in analogy with linear chains including the effects of dynamic dilution of the branches but neglecting the effects of branch point friction, in excellent agreement with the situation for linear polymers. Flow-birefringence measurements were performed in a Couette geometry, and the extracted steady-state shear and first normal stress differences were found to agree well with the mechanical data, but were limited to relatively low rates below the inverse stretch time of the backbone. Finally, the steady-state properties were found to be in good agreement with model predictions based on a nonlinear multimode tube model developed for linear polymers when the branches are treated as solvent.
Hayek, Mohamed
2016-04-01
This work develops a simple exact and explicit solution of the one-dimensional transient and nonlinear Richards' equation for soils in a special case of exponential water retention curve and power law hydraulic conductivity. The exact solution is obtained as traveling wave based on the approach proposed by Philip (1957, 1967) and adopted by Zlotnik et al. (2007). The obtained solution is novel, and it expresses explicitly the water content as function of the depth and time. It can be useful to model infiltration into semi-infinite soils with time-dependent boundary conditions and infiltration with constant boundary condition but space-dependent initial condition. A complete analytical inverse procedure based on the proposed analytical solution is presented which allows the estimation of hydraulic parameters. The proposed exact solution is also important for the verification of numerical schemes as well as for checking the implementation of time-dependent boundary conditions.
Exact Solutions for a Local Fractional DDE Associated with a Nonlinear Transmission Line
Aslan, İsmail
2016-09-01
Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions (kink/antikink solitons, singular, periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform.
Energy Technology Data Exchange (ETDEWEB)
Aslan, İsmail, E-mail: ismailaslan@iyte.edu.tr [Department of Mathematics, Izmir Institute of Technology, Urla, İzmir 35430 (Turkey)
2011-11-14
We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G{sup ′}/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before. -- Highlights: ► Discrete nonlinear Schrödinger equation with a saturable nonlinearity. ► We confirm that the model supports three types of solutions with arbitrary parameters. ► A new application of the (G{sup ′}/G)-expansion method presented.
Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher-Order Nonlinear Wave Equations
Directory of Open Access Journals (Sweden)
Lijun Zhang
2015-01-01
Full Text Available We study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a general sixth-order KdV equation.
Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
Yaşar, Emrullah; San, Sait; Özkan, Yeşim Sağlam
2016-01-01
In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.
Energy Technology Data Exchange (ETDEWEB)
Wang Qi [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China) and Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080 (China)]. E-mail: wangqi_dlut@yahoo.com.cn; Chen Yong [Nonlinear Science Center, Department of Mathematics, Ningbo University, Ningbo 315211 (China)
2007-01-15
With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.
Directory of Open Access Journals (Sweden)
Ping Liu
2015-08-01
Full Text Available The symmetry reduction equations, similarity solutions, sub-groups and exact solutions of the (3+1-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity (INHBV equations, which describe the atmospheric gravity waves, are researched in this paper. Calculation on symmetry shows that the equations are invariant under the Galilean transformations, scaling transformations, rotational transformations and space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+1-dimensional INHBV equations are proposed. Traveling wave solutions of the INHBV equations are demonstrated by means of symmetry method. The evolutions on the wind velocities and temperature perturbation are demonstrated by figures.
Liu, Ping; Zeng, Bao-Qing; Deng, Bo-Bo; Yang, Jian-Rong
2015-08-01
The symmetry reduction equations, similarity solutions, sub-groups and exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity (INHBV equations), which describe the atmospheric gravity waves, are researched in this paper. Calculation on symmetry shows that the equations are invariant under the Galilean transformations, scaling transformations, rotational transformations and space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHBV equations are proposed. Traveling wave solutions of the INHBV equations are demonstrated by means of symmetry method. The evolutions on the wind velocities and temperature perturbation are demonstrated by figures.
Institute of Scientific and Technical Information of China (English)
YANGYong; YANZhen－Ya
2002-01-01
In this letter the three-dimensional nonlinear Helmholtz equation is investigated.which describes electromagnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic function solutions are obtained,by using our extended Jacobian elliptic function expansion method.When the modulus m-→1 or 0,the corresponding solitary waves including bright solitons,dark solitons and new line solitons and singly periodic solutions can be also found.
Seven common errors in finding exact solutions of nonlinear differential equations
Kudryashov, Nikolai A.
2009-01-01
We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We s
Indian Academy of Sciences (India)
Ranjit Kumar
2012-09-01
Travelling and solitary wave solutions of certain coupled nonlinear diffusion-reaction equations have been constructed using the auxiliary equation method. These equations arise in a variety of contexts not only in biological, chemical and physical sciences but also in ecological and social sciences.
Institute of Scientific and Technical Information of China (English)
Lu Jun
2004-01-01
The stationary-state nonlinear Schr(o)dinger equation, which models the dilute-gas Bose-Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick.The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.
Group Classification and Exact Solutions of a Class of Variable Coefficient Nonlinear Wave Equations
Institute of Scientific and Technical Information of China (English)
HUANG Ding-Jiang; MEI Jian-Qin; ZHANG Hong-Qing
2009-01-01
Complete group classification of a class of variable coefficient (1 + 1)-dimensional wave equations is performed.The possible additional equivalence transformations between equations from the class under consideration and the conditional equivalence groups are also investigated. These allow simplification of the results of the classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of special forms of these equations via the classical Lie method. Nonclassical symmetries of the wave equations are discussed.
Exact travelling wave solutions for a class of nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Xie Fuding E-mail: xiefd@sohu.com; Gao Xiaoshan
2004-03-01
In this paper, the tanh-method is improved by means of a proper rational transformation based upon a coupled projective Riccati equations. The ansatz can be applied to find more and new exact solutions of the partial differential equations with the aid of symbolic computation system, Maple. We choose an example, which includes phi{sup 4} equation, Klein-Gordon equation, Duffing equation, Landau-Ginburg-Higgs equation and Sine-Gordon equation, to illustrate the method.
EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION
Institute of Scientific and Technical Information of China (English)
姚若侠; 李忠斌
2003-01-01
A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.
Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems
Directory of Open Access Journals (Sweden)
Oleksii Pliukhin
2015-10-01
Full Text Available A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, the solutions of some reaction–diffusion systems of the Lotka–Volterra type in an explicit form and satisfying Dirichlet boundary conditions are obtained. An biological interpretation is presented in order to show that two different types of interaction between biological species can be described.
Li, Biao; Chen, Yong; Zhang, Hongqing
2004-02-01
In this paper, by introducing some proper transformations, the applied range of the homogenous balance (HB) method is extended. With the help of Mathematica, we obtain three auto-Bäcklund transformations (BT) for the generalized Fithugh-Nagumo equation, the generalized Burgers-Fisher equation, the generalized Burgers-Huxley equation, respectively, by use of the extended HB method. From these BTs, some exact solutions for these equations are derived.
Vitanov, Nikolay K.
2011-03-01
We discuss the class of equations ∑i,j=0mAij(u){∂iu}/{∂ti}∂+∑k,l=0nBkl(u){∂ku}/{∂xk}∂=C(u) where Aij( u), Bkl( u) and C( u) are functions of u( x, t) as follows: (i) Aij, Bkl and C are polynomials of u; or (ii) Aij, Bkl and C can be reduced to polynomials of u by means of Taylor series for small values of u. For these two cases the above-mentioned class of equations consists of nonlinear PDEs with polynomial nonlinearities. We show that the modified method of simplest equation is powerful tool for obtaining exact traveling-wave solution of this class of equations. The balance equations for the sub-class of traveling-wave solutions of the investigated class of equations are obtained. We illustrate the method by obtaining exact traveling-wave solutions (i) of the Swift-Hohenberg equation and (ii) of the generalized Rayleigh equation for the cases when the extended tanh-equation or the equations of Bernoulli and Riccati are used as simplest equations.
Institute of Scientific and Technical Information of China (English)
Targut (O)zi(s); Imail Asian
2009-01-01
With the aid of symbolic computation system Mathematica, several explicit solutions for Fisher's equation and CKdV equation are constructed by utilizing an auxiliary equation method, the so called G'/G-expansion method, where the new and more general forms of solutions are also constructed. When the parameters are taken as special values, the previously known solutions are recovered.
Cherniha, Roman
2017-01-01
This book presents several fundamental results in solving nonlinear reaction-diffusion equations and systems using symmetry-based methods. Reaction-diffusion systems are fundamental modeling tools for mathematical biology with applications to ecology, population dynamics, pattern formation, morphogenesis, enzymatic reactions and chemotaxis. The book discusses the properties of nonlinear reaction-diffusion systems, which are relevant for biological applications, from the symmetry point of view, providing rigorous definitions and constructive algorithms to search for conditional symmetry (a nontrivial generalization of the well-known Lie symmetry) of nonlinear reaction-diffusion systems. In order to present applications to population dynamics, it focuses mainly on two- and three-component diffusive Lotka-Volterra systems. While it is primarily a valuable guide for researchers working with reaction-diffusion systems and those developing the theoretical aspects of conditional symmetry conception,...
Institute of Scientific and Technical Information of China (English)
YANG Yong; YAN Zhen-Ya
2002-01-01
In this letter the three-dimensional nonlinear Helmholtz equation is investigated, which describes electro-magnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic functionsolutions are obtained, by using our extended Jacobian elliptic function expansion method. When the modulus m → 1or0, the corresponding solitary waves including bright solitons, dark solitons and new line solitons and singly periodicsolutions can be also found.
Institute of Scientific and Technical Information of China (English)
JianlanHU; X.FENG; ZhiLi
2000-01-01
New exact traveling wave solutions are derived for the fifth order KdV type equations by using a delicate way of rank analysis two-step ansatz method. Solitary shallowwater waves described by the above equation are discussed.
On exact solutions of modified KdV-ZK equation
Directory of Open Access Journals (Sweden)
Syed Tauseef Mohyud-Din
2016-12-01
Full Text Available In this work, we established some exact particular solutions with parameters for Modified KdV-ZK Equation. The improved tanϕ(ξ2-expansion method is introduced to construct exact particular solutions of nonlinear evolution equations. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. The method appears to be effective mathematical tool for solving nonlinear evolution equations (NLEEs in applied mathematics and engineering.
Directory of Open Access Journals (Sweden)
ömer gözükızıl
2015-12-01
Full Text Available In this paper, the G'/G expansion method with the aid of computer algebraic system Maple is used for constructing exact travelling wave solutions and new kinds of solutions for the modified dispersive water wave equations, the Abrahams-Tsuneto reaction diffusion system, and for a class of reaction diffusion models. The method is straightforward and concise, and it be also applied to other nonlinear partial differential equations.
New exact wave solutions for Hirota equation
Indian Academy of Sciences (India)
M Eslami; M A Mirzazadeh; A Neirameh
2015-01-01
In this paper, we construct the topological or dark solitons of Hirota equation by using the first integral method. This approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solution is constructed through the established first integrals. This method is a powerful tool for searching exact travelling solutions of nonlinear partial differential equations (NPDEs) in mathematical physics.
Exact solutions of a two-dimensional cubic–quintic discrete nonlinear Schrödinger equation
DEFF Research Database (Denmark)
Khare, Avinash; Rasmussen, Kim Ø; Samuelsen, Mogens Rugholm
2011-01-01
We show that a two-dimensional generalized cubic–quintic Ablowitz–Ladik lattice admits periodic solutions that can be expressed in analytical form. The framework for the stability analysis of these solutions is developed and applied to reveal the intricate stability behavior of this nonlinear sys...
Energy Technology Data Exchange (ETDEWEB)
Li Wenting [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)], E-mail: lwt.wentinglee@yahoo.com.cn; Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2009-03-15
Based on symbolic computation and the idea of rational expansion method, a new generalized compound Riccati equations rational expansion method (GCRERE) is suggested to construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general complexiton solutions. The validity and reliability of the method is tested by its application to the (2+1)-dimensional Burgers equation. It is shown that more complexiton solutions can be found by this new method.
Exact solution to fractional logistic equation
West, Bruce J.
2015-07-01
The logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences. Herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time. The solution to the fractional logistic equation (FLE) is obtained using the Carleman embedding technique that allows the nonlinear equation to be replaced by an infinite-order set of linear equations, which we then solve exactly. The formal series expansion for the initial value solution of the FLE is shown to be expressed in terms of a series of weighted Mittag-Leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity. The numerical integration to the FLE provides an excellent fit to the analytic solution. We propose this approach as a general technique for solving a class of nonlinear fractional differential equations.
Akbar, M Ali; Ali, Norhashidah Hj Mohd; Mohyud-Din, Syed Tauseef
2013-01-01
The (G'/G)-expansion method is one of the most direct and effective method for obtaining exact solutions of nonlinear partial differential equations (PDEs). In the present article, we construct the exact traveling wave solutions of nonlinear evolution equations in mathematical physics via the (2 + 1)-dimensional breaking soliton equation by using two methods: namely, a further improved (G'/G)-expansion method, where G(ξ) satisfies the auxiliary ordinary differential equation (ODE) [G'(ξ)](2) = p G (2)(ξ) + q G (4)(ξ) + r G (6)(ξ); p, q and r are constants and the well known extended tanh-function method. We demonstrate, nevertheless some of the exact solutions bring out by these two methods are analogous, but they are not one and the same. It is worth mentioning that the first method has not been exercised anybody previously which gives further exact solutions than the second one. PACS numbers 02.30.Jr, 05.45.Yv, 02.30.Ik.
Chen Yong; Zhang Hong Qin
2003-01-01
Based on the idea of homogenous balance method and with the help of Mathematica, we obtain a new auto-Baecklund transformation for modified nonlinear dispersive equation mK(m,n). Then based on the Baecklund transformation, some solitary patterns solution for mK(m,n) equation are derived. In addition, we also obtain the general solutions for mK(n,n) in higher dimensional spatial domains, even in N dimensional space.
Institute of Scientific and Technical Information of China (English)
张卫国
2003-01-01
In this paper, we have obtained the bell-type and kink-type solitary wave solutions of the generalized symmetric regularized long-wave equations with high-order nonlinear terms by means of proper transformation and undetermined assumption method.
Aziz, Taha; Mahomed, F M
2014-01-01
In this communication, we utilize some basic symmetry reductions to transform the governing nonlinear partial differential equations arising in the study of third-grade fluid flows into ordinary differential equations. We obtain some simple closed-form steady-state solutions of these reduced equations. Our solutions are valid for the whole domain [0,∞) and also satisfy the physical boundary conditions. We also present the numerical solutions for some of the underlying equations. The graphs corresponding to the essential physical parameters of the flow are presented and discussed.
Exact solution for generalized pairing
Pan, Feng; J.P. Draayer
1997-01-01
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with some numerical examples.
Exact cosmological solutions for MOG
Energy Technology Data Exchange (ETDEWEB)
Roshan, Mahmood [Ferdowsi University of Mashhad, Department of Physics, P.O. Box 1436, Mashhad (Iran, Islamic Republic of)
2015-09-15
We find some new exact cosmological solutions for the covariant scalar-tensor-vector gravity theory, the so-called modified gravity (MOG). The exact solution of the vacuum field equations has been derived. Also, for non-vacuum cases we have found some exact solutions with the aid of the Noether symmetry approach. More specifically, the symmetry vector and also the Noether conserved quantity associated to the point-like Lagrangian of the theory have been found. Also we find the exact form of the generic vector field potential of this theory by considering the behavior of the relevant point-like Lagrangian under the infinitesimal generator of the Noether symmetry. Finally, we discuss the cosmological implications of the solutions. (orig.)
Santucci, F.; Santini, P. M.
2016-10-01
We study the generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if m(n-1)≤slant 2. Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [1], generalize those obtained in [2] for the dKP equation in n+1 dimensions with quadratic nonlinearity, and are obtained following the same strategy.
CSIR Research Space (South Africa)
Mhlongo, MD
2014-05-01
Full Text Available Solutions of Nonlinear Fin Problem for Steady Heat Transfer in Longitudinal Fin with Different Profiles M. D. Mhlongo1 and R. J. Moitsheki2 1 Defence, Peace, Safety and Security, Landward Sciences, Council for Scientific and Industrial Research, P.O. Box 395... efficiency are studied. 1. Introduction Heat transfer through extended surfaces has been studied quite extensively [1], perhaps because of its frequent applica- tions in engineering. Through the process of mathematical modeling, heat transfer problems...
Saha Ray, S.
2016-09-01
In this article, the Jacobi elliptic function method viz. the mixed dn-sn method has been presented for finding the travelling wave solutions of the Davey-Stewartson equations. As a result, some solitary wave solutions and doubly periodic solutions are obtained in terms of Jacobi elliptic functions. Moreover, solitary wave solutions are obtained as simple limits of doubly periodic functions. These solutions can be useful to explain some physical phenomena, viz. evolution of a three-dimensional wave packet on water of finite depth. The proposed Jacobi elliptic function method is efficient, powerful and can be used in order to establish newer exact solutions for other kinds of nonlinear fractional partial differential equations arising in mathematical physics.
Cooper, Fred; Khare, Avinash; Comech, Andrew; Mihaila, Bogdan; Dawson, John F.; Saxena, Avadh
2017-01-01
We discuss the stability properties of the solutions of the general nonlinear Schrödinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time ({ P }{ T }) symmetric superpotential W(x) that we considered earlier, Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation \\{{{i}} {\\partial }t+{\\partial }x2-{V}-(x)+| \\psi (x,t){| }2κ \\} \\psi (x,t)=0, for arbitrary nonlinearity parameter κ. We study the bound state solutions when {V}-(x) =(1/4-{b}2){\\text{sech}}2(x), which can be derived from two different superpotentials W(x), one of which is complex and { P }{ T } symmetric. Using Derrick's theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth b 2 of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov-Kolokolov (V-K) stability criterion. The numerical results of applying the V-K condition give the same answer for the domain of stability as the analytic result obtained from applying Derrick's theorem. Our main result is that for κ \\gt 2 a new regime of stability for the exact solutions appears as long as b\\gt {b}{{crit}}, where {b}{{crit}} is a function of the nonlinearity parameter κ. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for κ \\gt 2.
Directory of Open Access Journals (Sweden)
M. D. Mhlongo
2014-01-01
Full Text Available One-dimensional steady-state heat transfer in fins of different profiles is studied. The problem considered satisfies the Dirichlet boundary conditions at one end and the Neumann boundary conditions at the other. The thermal conductivity and heat coefficients are assumed to be temperature dependent, which makes the resulting differential equation highly nonlinear. Classical Lie point symmetry methods are employed, and some reductions are performed. Some invariant solutions are constructed. The effects of thermogeometric fin parameter, the exponent on temperature, and the fin efficiency are studied.
Exact solutions to the generalized Lienard equation and its applications
Indian Academy of Sciences (India)
Yang Xian-Lin; Tang Jia-Shi
2008-12-01
Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein–Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov–Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.
New Approach to Find Exact Solutions to Classical Boussinesq System
Institute of Scientific and Technical Information of China (English)
ZHI Hong-Yan; ZHAO Xue-Qin; ZHANG Hong-Qing
2005-01-01
In this paper, based on a new system of three Riccati equations, we give a new method to construct more new exact solutions of nonlinear differential equations in mathematical physics. The classical Boussinesq system is chosen to illustrate our method. As a consequence, more families of new exact solutions are obtained, which include solitary wave solutions and periodic solutions.
Exact analytical solutions for ADAFs
Habibi, Asiyeh; Shadmehri, Mohsen
2016-01-01
We obtain two-dimensional exact analytic solutions for the structure of the hot accretion flows without wind. We assume that the only non-zero component of the stress tensor is $T_{r\\varphi}$. Furthermore we assume that the value of viscosity coefficient $\\alpha$ varies with $\\theta$. We find radially self-similar solutions and compare them with the numerical and the analytical solutions already studied in the literature. The no-wind solution obtained in this paper may be applied to the nuclei of some cool-core clusters.
A Simple Method to Obtain Exact Soliton Solutions for a Nonlinear Equation in a Loss Fibre System
Institute of Scientific and Technical Information of China (English)
YANGXiao－Xue; WUYing; 等
2002-01-01
We show that the nonlinear equation governing wave propagation in a loss fibre system considered by Nakkerian in J.Phys.A34(2001) 5111 can be brought into the standard nonlinear schroedinger equation by a simple transformation.
New Exact Travelling Wave Solutions to Kundu Equation
Institute of Scientific and Technical Information of China (English)
HUANG Ding-Jiang; LI De-Sheng; ZHANG Hong-Qing
2005-01-01
Based on a first-order nonlinear ordinary differential equation with six-degree nonlinear term, we first present a new auxiliary equation expansion method and its algorithm. Being concise and straightforward, the method is applied to the Kundu equation. As a result, some new exact travelling wave solutions are obtained, which include bright and dark solitary wave solutions, triangular periodic wave solutions, and singular solutions. This algorithm can also be applied to other nonlinear evolution equations in mathematical physics.
Sinha, Debdeep; Ghosh, Pijush K
2015-04-01
A class of nonlocal nonlinear Schrödinger equations (NLSEs) is considered in an external potential with a space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable nonlocal NLSE with self-induced potential that is parity-time-symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or nonautonomous nonlocal NLSE by using similarity transformation, and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the nonlocal NLSE without the external potential and a (d+1)-dimensional generalization of it admits all the symmetries of the (d+1)-dimensional Schrödinger group. The conserved Noether charges associated with the time translation, dilatation, and special conformal transformation are shown to be real-valued in spite of being non-Hermitian. Finally, the dynamics of different moments are studied with an exact description of the time evolution of the "pseudowidth" of the wave packet for the special case in which the system admits a O(2,1) conformal symmetry.
Sinha, Debdeep; Ghosh, Pijush K.
2015-04-01
A class of nonlocal nonlinear Schrödinger equations (NLSEs) is considered in an external potential with a space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable nonlocal NLSE with self-induced potential that is parity-time-symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or nonautonomous nonlocal NLSE by using similarity transformation, and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the nonlocal NLSE without the external potential and a (d +1 )-dimensional generalization of it admits all the symmetries of the (d +1 )-dimensional Schrödinger group. The conserved Noether charges associated with the time translation, dilatation, and special conformal transformation are shown to be real-valued in spite of being non-Hermitian. Finally, the dynamics of different moments are studied with an exact description of the time evolution of the "pseudowidth" of the wave packet for the special case in which the system admits a O (2 ,1 ) conformal symmetry.
Institute of Scientific and Technical Information of China (English)
GE Jian-Ya; WANG Rui-Min; DAI Chao-Qing; ZHANG Jie-Fang
2006-01-01
In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schr(o)dinger equation with variable-coefficient. These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time. Six figures are given to illustrate some features of these solutions. The method can be applied to other nonlinear evolution equations in mathematical physics.
Mahabaleshwar, U. S.; Nagaraju, K. R.; Vinay Kumar, P. N.; Baleanu, Dumitru; Lorenzini, Giulio
2017-03-01
In this paper, we investigate the theoretical analysis for the unsteady magnetohydrodynamic laminar boundary layer flow due to impulsively stretching sheet. The third-order highly nonlinear partial differential equation modeling the unsteady boundary layer flow brought on by an impulsively stretching flat sheet was solved by applying Adomian decomposition method and Pade approximants. The exact analytical solution so obtained is in terms of rapidly converging power series and each of the variants are easily computable. Variations in parameters such as mass transfer (suction/injection) and Chandrasekhar number on the velocity are observed by plotting the graphs. This particular problem is technically sound and has got applications in expulsion process and related process in fluid dynamics problems.
Mahabaleshwar, U. S.; Nagaraju, K. R.; Vinay Kumar, P. N.; Baleanu, Dumitru; Lorenzini, Giulio
2016-12-01
In this paper, we investigate the theoretical analysis for the unsteady magnetohydrodynamic laminar boundary layer flow due to impulsively stretching sheet. The third-order highly nonlinear partial differential equation modeling the unsteady boundary layer flow brought on by an impulsively stretching flat sheet was solved by applying Adomian decomposition method and Pade approximants. The exact analytical solution so obtained is in terms of rapidly converging power series and each of the variants are easily computable. Variations in parameters such as mass transfer (suction/injection) and Chandrasekhar number on the velocity are observed by plotting the graphs. This particular problem is technically sound and has got applications in expulsion process and related process in fluid dynamics problems.
Exact solutions of (3 + 1)-dimensional stochastic Burgers equation
Energy Technology Data Exchange (ETDEWEB)
Wang Tieying [Department of Applied Mathematics and Physics, Dalian Nationalities for University, Dalian 116600 (China)]. E-mail: wangty@dlnu.edu.cn; Ren Yonghong [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China); Zhao Yali [Department of Mathematics, Chaoyang Teachers College, Chaoyang 122000 (China)
2006-08-15
A generalized tan h function method is used for constructing exact travelling wave solutions of nonlinear stochastic partial differential equations. The main idea of this method is to take full advantage of the Riccati equation, which has more exact solutions. More Wick-type stochastic multiple soliton-like solutions and triangular periodic solutions are obtained for the (3 + 1)-dimensional Wick-type stochastic Burgers equation via Hermite transformation.
Exact Nonlinear Internal Equatorial Waves in the f-plane
Hsu, Hung-Chu
2016-07-01
We present an explicit exact solution of the nonlinear governing equations for internal geophysical water waves propagating westward above the thermocline in the f-plane approximation near the equator. Moreover, the mass transport velocity induced by this internal equatorial wave is eastward and a westward current occurs in the transition zone between the great depth where the water is still and the thermocline.
Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation
Directory of Open Access Journals (Sweden)
Xun Liu
2010-01-01
Full Text Available We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.
An exact solution of Haugan's binary pulsar equation of motion
Weinstein, M.; Mor, A.
1988-05-01
In his work on the post-Newtonian arrival-time analysis for a pulsary binary system, Haugan (1985) derived and integrated the two-body equation of the motion of the pulsar. The purpose of the present study is to show that there is an exact solution to this nonlinear equation, without any need of far-reaching assumptions and neglected nonlinear terms.
Exact Solitary Wave Solution in the ZK-BBM Equation
Directory of Open Access Journals (Sweden)
Juan Zhao
2014-01-01
Full Text Available The traveling wave solution for the ZK-BBM equation is considered, which is governed by a nonlinear ODE system. The bifurcation structure of fixed points and bifurcation phase portraits with respect to the wave speed c are analyzed by using the dynamical system theory. Furthermore, the exact solutions of the homoclinic orbits for the nonlinear ODE system are obtained which corresponds to the solitary wave solution curve of the ZK-BBM equation.
El-Mistikawy, Tarek M. A.
2011-12-01
In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877-888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.
Energy Technology Data Exchange (ETDEWEB)
Chen Yong E-mail: chenyong@dlut.edu.cn; Li Biao E-mail: libiao@dlut.edu.cn
2004-03-01
Applying the improved generalized method, which is a direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear partial differential equations and implemented in a computer algebraic system, we consider the KdV-type equations and KdV-Burgers-type equations with nonlinear terms of any order. As a result, we can not only successfully recover the previously known travelling wave solutions found by existing various tanh methods and other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shaped solitons, bell-shaped solitons, singular solitons and periodic solutions.
Exact Solutions in Modified Gravity Models
Directory of Open Access Journals (Sweden)
Valery V. Obukhov
2012-06-01
Full Text Available We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned.
Exact Solutions in Modified Gravity Models
Makarenko, Andrey N
2012-01-01
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned.
EXACT SOLUTIONS OF SOME FIFTH－ORDER NONLINEAR EQUATIONS%某些五阶非线性方程的精确解
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
To solve the nonlinear partial differential equations is changed into solving some algebraic equations by using the function U to be expressed as linear independent functions. The new soliton and periodic solutions of some fifth-order nonlinear partial differential equations are obtained.
Exact Solution and Exotic Fluid in Cosmology
Directory of Open Access Journals (Sweden)
Phillial Oh
2012-09-01
Full Text Available We investigate cosmological consequences of nonlinear sigma model coupled with a cosmological fluid which satisfies the continuity equation. The target space action is of the de Sitter type and is composed of four scalar fields. The potential which is a function of only one of the scalar fields is also introduced. We perform a general analysis of the ensuing cosmological equations and give various critical points and their properties. Then, we show that the model exhibits an exact cosmological solution which yields a transition from matter domination into dark energy epoch and compare it with the Λ-CDM behavior. Especially, we calculate the age of the Universe and show that it is consistent with the observational value if the equation of the state ωf of the cosmological fluid is within the range of 0.13 < ωf < 0.22. Some implication of this result is also discussed.
Exact Wave Solutions for the High-order Dispersive Nonlinear Schr odinger Equation%高阶色散非线性薛定谔方程的精确波解
Institute of Scientific and Technical Information of China (English)
施业琼; 韩松
2009-01-01
利用行波约化方法,研究了用于描述飞秒光脉冲传输的高阶色散非线性薛定谔方程.通过借助一个新的高阶辅助方程的解,取得了该方程不同类型的包络型的孤波解、扭结波解、周期波解及奇异波解.%By using the traveling wave reduction method, a new auxiliary differential equation and its solutions are used to construct the exact traveling wave solutions, some new exact solutions for the high-order dispersive nonlinear Schrodinger equation are obtained which include bell profile solutions, kink profile solutions, triangular periodic solutions and singular solutions. This equation is used to describe the propagation of femtosecond optical pulses in fibers.
Exact Solutions for Einstein's Hyperbolic Geometric Flow
Institute of Scientific and Technical Information of China (English)
HE Chun-Lei
2008-01-01
In this paper we investigate the Einstein's hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein's hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.
非线性偏微分方程的约化和精确解%REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
叶彩儿; 潘祖梁
2003-01-01
Nonlinear partial differetial equation(NLPDE) is converted into ordinary differentialequation (ODE) via a new ansatzUsing undetermined function method ,the ODE obtained aboveis replaced by a set of algebraic equations which are solved out with the aid of MathematicaTheexact solutions and solitary solutions of NLPDE are obtained.
New exact solutions in standard inflationary models
Chervon, S V; Shchigolev, V K
1997-01-01
The exact solutions in the standard inflationary model based on the self-interacting scalar field minimally coupled to gravity are considered. The shape's freedom of the self-interacting potential $V(\\phi)$ is postulated to obtain a new set of the exact solutions in the framework of Friedmann-Robertson-Walker Universes. The general solution was found in the case of power law inflation. We obtained new solutions and compared them with obtained ones earlir for the exponential type inflation.
New exact solutions to some difference differential equations
Institute of Scientific and Technical Information of China (English)
Wang Zhen; Zhang Hong-Qing
2006-01-01
In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.
An exactly solvable three-dimensional nonlinear quantum oscillator
Energy Technology Data Exchange (ETDEWEB)
Schulze-Halberg, A. [Department of Mathematics and Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States); Morris, J. R. [Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States)
2013-11-15
Exact analytical, closed-form solutions, expressed in terms of special functions, are presented for the case of a three-dimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding one-dimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodrigues-type of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the s-states of the system, the radial equation accepts solutions that have been recently found for the one-dimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.
Generating exact solutions to Einstein's equation using linearized approximations
Harte, Abraham I.; Vines, Justin
2016-10-01
We show that certain solutions to the linearized Einstein equation can—by the application of a particular type of linearized gauge transformation—be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.
Generating exact solutions to Einstein's equation using linearized approximations
Harte, Abraham I
2016-01-01
We show that certain solutions to the linearized Einstein equation can---by the application of a particular type of linearized gauge transformation---be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.
Exact solutions in three-dimensional gravity
Garcia-Diaz, Alberto A
2017-01-01
A self-contained text, systematically presenting the determination and classification of exact solutions in three-dimensional Einstein gravity. This book explores the theoretical framework and general physical and geometrical characteristics of each class of solutions, and includes information on the researchers responsible for their discovery. Beginning with the physical character of the solutions, these are identified and ordered on the basis of their geometrical invariant properties, symmetries, and algebraic classifications, or from the standpoint of their physical nature, for example electrodynamic fields, fluid, scalar field, or dilaton. Consequently, this text serves as a thorough catalogue on 2+1 exact solutions to the Einstein equations coupled to matter and fields, and on vacuum solutions of topologically massive gravity with a cosmological constant. The solutions are also examined from different perspectives, enabling a conceptual bridge between exact solutions of three- and four-dimensional gravit...
Instability and exact multiplicity of solutions of semilinear equations
Directory of Open Access Journals (Sweden)
Philip Korman
2000-10-01
Full Text Available For a class of two-point boundary-value problems we use bifurcation theory to show that a solution is unstable under a simple, geometric in nature, assumption on the non-linear term. As an application we obtain some new exact multiplicity results.
Exact solutions of the time-fractional Fisher equation by using modified trial equation method
Tandogan, Yusuf Ali; Bildik, Necdet
2016-06-01
In this study, modified trial equation method has been proposed to obtain precise solutions of nonlinear fractional differential equation. Using the modified test equation method, we obtained some new exact solutions of the time fractional nonlinear Fisher equation. The obtained results are classified as a soliton solution, singular solutions, rational function solutions and periodic solutions.
Exact control of parity-time symmetry in periodically modulated nonlinear optical couplers
Yang, Baiyuan; Hu, QiangLin; Yu, XiaoGuang
2016-01-01
We propose a mechanism for realization of exact control of parity-time (PT) symmetry by using a periodically modulated nonlinear optical coupler with balanced gain and loss. It is shown that for certain appropriately chosen values of the modulation parameters, we can construct a family of exact analytical solutions for the two-mode equations describing the dynamics of such nonlinear couplers. These exact solutions give explicit examples that allow us to precisely manipulate the system from nonlinearity-induced symmetry breaking to PT symmetry, thus providing an analytical approach to the all-optical signal control in nonlinear PT-symmetric structures.
Institute of Scientific and Technical Information of China (English)
王云虎; 陈勇
2011-01-01
In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.
On exact solutions of the Bogoyavlenskii equation
Indian Academy of Sciences (India)
Yan-Ze Peng; Ming Shen
2006-09-01
Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a phase-shift. The dromion-like structures with elastic and nonelastic interactions are found.
On the exact controllability of a nonlinear stochastic heat equation
Directory of Open Access Journals (Sweden)
Bui An Ton
2006-01-01
Full Text Available The exact controllability of a nonlinear stochastic heat equation with null Dirichlet boundary conditions, nonzero initial and target values, and an interior control is established.
Constructing exact solutions to discrete systems with the trial function method
Institute of Scientific and Technical Information of China (English)
Taogetusang Sirendaoerji
2008-01-01
Based on the homogenous balance method and the trial function method, several trial function methods composed of exponential functions are proposed and applied to nonlinear discrete systems. With the help of symbolic computation system, the new exact solitary wave solutions to discrete nonlinear mKdV lattice equation, discrete nonlinear (2+1) dimensional Toda lattice equation, Ablowitz-Ladik-lattice system are constructed.The method is of significance to seek exact solitary wave solutions to other nonlinear discrete systems.
Exact Solutions in Nonlocal Linear Models
Vernov, S. Yu.
2008-01-01
A general class of cosmological models driven by a nonlocal scalar field inspired by the string field theory is studied. Using the fact that the considering linear nonlocal model is equivalent to an infinite number of local models we have found an exact special solution of the nonlocal Friedmann equations. This solution describes a monotonically increasing Universe with the phantom dark energy.
Classes of exact Einstein Maxwell solutions
Komathiraj, K.; Maharaj, S. D.
2007-12-01
We find new classes of exact solutions to the Einstein Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.
EXACT TRAVELLING WAVE SOLUTIONS TO BBM EQUATION
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Abundant new travelling wave solutions to the BBM (Benjamin-Bona-Mahoni) equation are obtained by the generalized Jacobian elliptic function method. This method can be applied to other nonlinear evolution equations.
Vitanov, Nikolay K.
2010-08-01
We search for traveling-wave solutions of the class of equations ∑p=1N1αp{∂pQ}/{∂tp}+∑q=1N2βq{∂qQ}/{∂xq}+∑m=1MμmQm=0 where αp,βq and μm are parameters. We obtain such solutions by the method of simplest equation for the cases when the simplest equation is the the equation of Bernoulli or the equation of Riccati. We modify the methodology of the simplest equation of Kudryashov as follows. Kudryashov uses the first step of the test for Painleve property in order to determine the size of the solution of the studied PDE. We divide the studied PDEs in two parts: part A which contains the derivatives, and part B which contains the rest of the equation. The application of the ansatz or the extended ansatz of Kudryashov, transforms part A and part B to two polynomials. We balance the highest powers of the polynomials for the parts A and B and, thus obtain a balance equation which depends on the kind of the used simplest equation. The balance equations are connected to nonlinear algebraic systems of relationships among the parameters of the equations and the parameters of the solution. On the basis of these systems, we obtain numerous solutions of the studied class of equations.
Exact Vacuum Solutions to the Einstein Equation
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations,which are much convenient for the resolution.
EXACT SOLUTIONS OF A DIPOLAR FLUID FLOW
Institute of Scientific and Technical Information of China (English)
T. HAYAT
2003-01-01
Exact solutions for three canonical flow problems of a dipolar fluid are obtained: (i)The flow of a dipolar fluid due to a suddenly accelerated plate, (ii) The flow generated by periodic oscillation of a plate, (iii) The flow due to plate oscillation in the presence of a transverse magnetic field. The solutions of some interesting flows caused by an arbitrary velocity of the plate and of certain special oscillations are also obtained.
Solutions of some class of nonlinear PDEs in mathematical physics
Directory of Open Access Journals (Sweden)
Shoukry El-Ganaini
2016-04-01
As a result, exact traveling wave solutions involving parameters have been obtained for the considered nonlinear equations in a concise manner. When these parameters are chosen as special values, the solitary wave solutions are derived. It is shown that the proposed technique provides a more powerful mathematical tool for constructing exact solutions for a broad variety of nonlinear PDEs in mathematical physics.
Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach
Institute of Scientific and Technical Information of China (English)
DAI Chao-Qing; ZHANG Jie-Fang
2006-01-01
In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.
Exact cosmological solutions from Hojman conservation quantities
Energy Technology Data Exchange (ETDEWEB)
Capozziello, Salvatore, E-mail: capozzie@na.infn.it [Dipartimento di Fisica, Università di Napoli “Federico II”, Napoli (Italy); INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli (Italy); Roshan, Mahmood, E-mail: rowshan@alumni.ut.ac.ir [Department of Physics, Ferdowsi University of Mashhad, P.O. Box 1436, Mashhad (Iran, Islamic Republic of)
2013-10-07
We present a new approach to find exact solutions for cosmological models. By requiring the existence of a symmetry transformation vector for the equations of motion of the given cosmological model (without using either Lagrangian or Hamiltonian), one can find corresponding Hojman conserved quantities. With the help of these conserved quantities, the analysis of the cosmological model can be simplified. In the case of quintessence scalar–tensor models, we show that the Hojman conserved quantities exist for a wide range of V(ϕ)-potentials and allow to find exact solutions for the cosmic scale factor and the scalar field. Finally, we investigate the general cosmological behavior of solutions by adopting a phase-space view.
Exact cosmological solutions from Hojman conservation quantities
Capozziello, Salvatore
2013-01-01
We present a new approach to find exact solutions for cosmological models. By requiring the existence of a symmetry transformation vector for the equations of motion of the given cosmological model (without using either Lagrangian or Hamiltonian), one can find corresponding Hojman conserved quantities. With the help of these conserved quantities, the analysis of the cosmological model can be simplified. In the case of quintessence scalar-tensor models, we show that the Hojman conserved quantities exist for a wide range of V(\\phi)-potentials and allow to find exact solutions for the cosmic scale factor and the scalar field. Finally, we investigate the general cosmological behavior of solutions by adopting a phase-space view.
Exact Traveling Wave Solutions for Wick-Type Stochastic Schamel KdV Equation
Directory of Open Access Journals (Sweden)
Hossam A. Ghany
2014-01-01
Full Text Available F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. By means of Hermite transform, inverse Hermite transform, and white noise analysis, the variable coefficients and Wick-type stochastic Schamel KdV equations are completely described. Abundant exact traveling wave solutions for variable coefficients Schamel KdV equations are given. These solutions include exact stochastic Jacobi elliptic functions, trigonometric functions, and hyperbolic functions solutions.
New explicit and exact solutions of the Benney-Kawahara-Lin equation
Institute of Scientific and Technical Information of China (English)
Xie Yuan-Xi
2009-01-01
In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.
Institute of Scientific and Technical Information of China (English)
张少武; 易林
2009-01-01
在获得一个含变化3.5阶非线性、弱非局域性、增益及非线性增益的广义薛定谔方程的自相似解的基础上,采用数值方法研究了解的稳定性.结果表明,在同时具有或没有非局域性和5阶非线性的介质中可以形成与传播自相似波;而且当相位参数远离士～1/2时,非局域度和累积衍射将极大影响自相似波的稳定性.%Exact self-similar solution of a generalized nonlinear Schrodinger equation with varying cubic-quintic nonlinearity, weakly nonlocality, gain and nonlinear gain was obtained. The stability of the solution was studied numerically. The results show that the self-similar solitary wave can exist and propagate in the media with or without both nonlocality and quintic nonlinearity, and that the stability of the self-similar solitary wave is drastically influenced by the degree of nonlocality and the cumulative diffraction under the condition that the phase parameter is far from ±√2
Stokes Waves Revisited: Exact Solutions in the Asymptotic Limit
Davies, Megan
2016-01-01
Stokes perturbative solution of the nonlinear (boundary value dependent) surface gravity wave problem is known to provide results of reasonable accuracy to engineers in estimating the phase speed and amplitudes of such nonlinear waves. The weakling in this structure though is the presence of aperiodic secular variation in the solution that does not agree with the known periodic propagation of surface waves. This has historically necessitated increasingly higher ordered (perturbative) approximations in the representation of the velocity profile. The present article ameliorates this long standing theoretical insufficiency by invoking a compact exact $n$-ordered solution in the asymptotic infinite depth limit, primarily based on a representation structured around the third ordered perturbative solution, that leads to a seamless extension to higher order (e.g. fifth order) forms existing in the literature. The result from this study is expected to improve phenomenological engineering estimates, now that any desir...
Exact solution of phantom dark energy model
Institute of Scientific and Technical Information of China (English)
Wang Wen-Fu; Shui Zheng-Wei; Tang Bin
2010-01-01
We investigate the phantom dark energy model derived from the scalar field with a negative kinetic term. By assuming a particular relation between the time derivative of the phantom field and the Hubble function, an exact solution of the model is constructed. Absence of the 'big rip' singularity is shown explicitly. We then derive special features of phantom dark energy model and show that its predictions are consistent with all astrophysical observations.
Exact solutions for Weyl fermions with gravity
Energy Technology Data Exchange (ETDEWEB)
Cianci, Roberto; Fabbri, Luca; Vignolo, Stefano [Universita di Genova, DIME Sez. Metodi e Modelli Matematici, Genoa (Italy)
2015-10-15
We consider the single-handed spinor field in interaction with its own gravitational field described by the set of field equations given by the Weyl field equations written in terms of derivatives that are covariant with respect to the gravitational connection plus Einstein field equations soured with the energy tensor of the spinor: for the Weyl spinor and the ensuing spacetime of Weyl-Lewis-Papapetrou structure, we find all exact solutions. The obtained solution for the metric tensor is that of a PP-wave spacetime, while the spinor field is a flag-dipole. (orig.)
Nonuniform Braneworld Stars: AN Exact Solution
Ovalle, J.
In this paper the first exact interior solution to Einstein's field equations for a static and nonuniform braneworld star with local and nonlocal bulk terms is presented. It is shown that the bulk Weyl scalar U(r) is always negative inside the stellar distribution, and in consequence it reduces both the effective density and the effective pressure. It is found that the anisotropy generated by bulk gravity effect has an acceptable physical behavior inside the distribution. Using a Reissner-Nördstrom-like exterior solution, the effects of bulk gravity on pressure and density are found through matching conditions.
Exact Controllability for a Class of Nonlinear Evolution Control Systems
Institute of Scientific and Technical Information of China (English)
L¨u Yue; Li Yong
2015-01-01
In this paper, we study the exact controllability of the nonlinear control systems. The controllability results by using the monotone operator theory are es-tablished. No compactness assumptions are imposed in the main results.
Exact traveling soliton solutions for the scalar Qiao equation
Abdoulkary, Saïdou; Aboubakar, Mahamoudou; Aboukar; Mohamadou, Alidou; Beda, Tibi
2015-01-01
We investigate exact traveling wave solutions of the scalar Qiao equation proposed by Li and Qiao (2010 J. Math. Phys. 51 042703) using the generalized auxiliary equation method. This equation is known to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation for a fixed value of k. Symmetries of the scalar Qiao equation and its solutions are also considered. The obtained solutions include kink and antikink solitons, bright and dark solitons, singular solutions and exponential solutions. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics. This work could be also relevant for numerical studies of the scalar Qiao equation.
The exact fundamental solution for the Benes tracking problem
Balaji, Bhashyam
2009-05-01
The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.
Exact periodic wave and soliton solutions in two-component Bose-Einstein condensates
Institute of Scientific and Technical Information of China (English)
Li Hua-Mei
2007-01-01
We present several families of exact solutions to a system of coupled nonlinear Schr(o)dinger equations. The model describes a binary mixture of two Bose-Einstein condensates in a magnetic trap potential. Using a mapping deformation method, we find exact periodic wave and soliton solutions, including bright and dark soliton pairs.
New exact travelling wave solutions of bidirectional wave equations
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2011-06-01
The surface water waves in a water tunnel can be described by systems of the form [Bona and Chen, Physica D116, 191 (1998)] \\begin{equation*} \\begin{cases} v_t + u_x + (uv)_x + au_{x x x} − bv_{x x t} = 0,\\\\ u_t + v_x + u u_x + cv_{x x x} − d u_{x x t} = 0, \\end{cases} \\tag{1} \\end{equation*} where , , and d are real constants. In general, the exact travelling wave solutions will be helpful in the theoretical and numerical study of the nonlinear evolution systems. In this paper, we obtain exact travelling wave solutions of system (1) using the modiﬁed tanh–coth function method with computerized symbolic computation.
NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD
Institute of Scientific and Technical Information of China (English)
Liu Zhifang; Zhang Shanyuan
2006-01-01
A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.
Submerged Landau jet: exact solutions, their meaning and application
Energy Technology Data Exchange (ETDEWEB)
Broman, Goran I [Blekinge Institute of Technology, Karlskrona (Sweden); Rudenko, Oleg V [M. V. Lomonosov Moscow State University, Faculty of Physics, Moscow (Russian Federation)
2010-01-31
Exact hydrodynamic solutions generalizing the Landau submerged jet solution are reviewed. It is shown how exact inviscid solutions can be obtained and how boundary layer viscosity can be included by introducing parabolic coordinates. The use of exact solutions in applied hydrodynamics and acoustics is discussed. A historical perspective on the discovery of a class of exact solutions and on the analysis of their physical meaning is presented. (methodological notes)
Strong nonlinear oscillators analytical solutions
Cveticanin, Livija
2017-01-01
This book outlines an analytical solution procedure of the pure nonlinear oscillator system, offering a solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter. Includes exercises.
Extension of Variable Separable Solutions for Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
JIA Hua-Bing; ZHANG Shun-Li; XU Wei; ZHU Xiao-Ning; WANG Yong-Mao; LOU Sen-Yue
2008-01-01
We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separablecation, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.
A method for constructing exact solutions and application to Benjamin Ono equation
Institute of Scientific and Technical Information of China (English)
Wang Zhen; Li De-Sheng; Lu Hui-Fang; Zhang Hong-Qing
2005-01-01
By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.
Exact Spherically Symmetric Solutions in Massive Gravity
Berezhiani, Z; Nesti, F; Pilo, L
2008-01-01
A phase of massive gravity free from pathologies can be obtained by coupling the metric to an additional spin-two field. We study the gravitational field produced by a static spherically symmetric body, by finding the exact solution that generalizes the Schwarzschild metric to the case of massive gravity. Besides the usual 1/r term, the main effects of the new spin-two field are a shift of the total mass of the body and the presence of a new power-like term, with sizes determined by the mass and the shape (the radius) of the source. These modifications, being source dependent, give rise to a dynamical violation of the Strong Equivalence Principle. Depending on the details of the coupling of the new field, the power-like term may dominate at large distances or even in the ultraviolet. The effect persists also when the dynamics of the extra field is decoupled.
Kundu, Anjan
2016-12-01
Integrable quantum field models are known to exist mostly in one space-dimension. Exploiting the concept of multi-time in integrable systems and a Lax matrix of higher scaling order, we construct a novel quantum field model in quasi-two dimensions involving interacting fields. The Yang-Baxter integrability is proved for the model by finding a new kind of commutation rule for its basic fields, representing nonstandard scalar fields along the transverse direction. In spite of a close link with the quantum Landau-Lifshitz equation, the present model differs widely from it, in its content and the result obtained. Using further the algebraic Bethe ansatz we solve exactly the eigenvalue problem of this quantum field model for all its higher conserved operators. The idea presented here should instigate the construction of a novel class of integrable field and lattice models and exploration of a new type of underlying algebras.
Energy Technology Data Exchange (ETDEWEB)
Kundu, Anjan, E-mail: anjan.kundu@saha.ac.in
2016-12-15
Integrable quantum field models are known to exist mostly in one space-dimension. Exploiting the concept of multi-time in integrable systems and a Lax matrix of higher scaling order, we construct a novel quantum field model in quasi-two dimensions involving interacting fields. The Yang–Baxter integrability is proved for the model by finding a new kind of commutation rule for its basic fields, representing nonstandard scalar fields along the transverse direction. In spite of a close link with the quantum Landau–Lifshitz equation, the present model differs widely from it, in its content and the result obtained. Using further the algebraic Bethe ansatz we solve exactly the eigenvalue problem of this quantum field model for all its higher conserved operators. The idea presented here should instigate the construction of a novel class of integrable field and lattice models and exploration of a new type of underlying algebras.
Institute of Scientific and Technical Information of China (English)
王俊杰; 王连堂; 杨宽德
2012-01-01
对具有任意阶非线性薛定谔方程的行波解进行了研究.给出具有任意阶非线性薛定谔方程的精确解.利用行波变换和辅助函数法把具有任意阶非线性薛定谔方程最终转化为一个非线性常微分方程的解,通过对这个微分方程的研究可以得到具有任意阶非线性薛定谔方程的行波解.具有任意阶非线性薛定谔方程存在的孤立波解、三角孤立波解、扭孤立波解、Jaoobi椭圆函数解.得到了具有任意阶非线性薛定谔方程新的Jacobi椭圆函数解.%In this paper, traveling wave solutions of the generalized derivative Schrodinger equation with any order are studied. We give exact solutions of the generalized derivative Schrodinger equation with any order. A auxiliary equation expansion method and its algorithm is proposed by studying a second order nonlinear ordinary differential equation. The method is applied to the generalized derivative Schrodinger equation with any order. The generalized derivative Schrodinger equation with any order exist singular solutions, triangular periodic wave solution and Jacobian elliptic function solutions. We give the new Jacobian elliptic function solutions of the generalized derivative Schrodinger e-quation with any order.
AUTO-DARBOUX TRANSFORMATION AND EXACT SOLUTIONS OF THE BRUSSELATOR REACTION DIFFUSION MODEL
Institute of Scientific and Technical Information of China (English)
闫振亚; 张鸿庆
2001-01-01
Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain some authors' results known.Secondly, by using a series of transformations, the model is reduced into a nonlinear reaction diffusion equation and then through using sine- cosine method, more exact solutions are found which contain soliton solutions.
Exact controllability for a nonlinear stochastic wave equation
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available The exact controllability for a semilinear stochastic wave equation with a boundary control is established. The target and initial spaces are L 2 ( G × H −1 ( G with G being a bounded open subset of R 3 and the nonlinear terms having at most a linear growth.
ABUNDANT EXACT SOLUTION STRUCTURES OF THE NIZHNIK-NOVIKOV-VESELOV EQUATION
Institute of Scientific and Technical Information of China (English)
张解放
2001-01-01
Using the extended homogeneous balance method, we have obtained abundant exact solution structures of a (2+ 1)-dimensional integrable model, the Nizhnik-Novikov-Veselov equation. By means of leading order terms analysis,the nonlinear transformations of the Nizhnik-Novikov-Veselov equation are given first, and then some special types of single solitary wave solution and multisoliton-like solutions are constructed.
Exact solutions and conservation laws for a generalized improved Boussinesq equation
Motsepa, Tanki; Khalique, Chaudry Masood
2016-06-01
In this paper we study a nonlinear generalized improved Boussinesq equation, which describes nonlinear dispersive wave phenomena. Exact solutions are derived by using the Lie symmetry analysis and the simplest equation methods. Moreover, conservation laws are constructed by using the multiplier method.
Directory of Open Access Journals (Sweden)
Özkan Güner
2014-01-01
Full Text Available We apply the functional variable method, exp-function method, and (G′/G-expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.
Institute of Scientific and Technical Information of China (English)
XIAZhi
2004-01-01
Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.
Some new exact solutions to the Burgers-Fisher equation and generalized Burgers-Fisher equation
Institute of Scientific and Technical Information of China (English)
Jiang Lu; Guo Yu-Cui; Xu Shu-Jiang
2007-01-01
Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular solitary wave solutions and some more complex solutions whose figures are given in the article. The result shows that the first integral method is one of the most effective approaches to obtain the solutions of the nonlinear partial differential equations.
Strongly nonlinear oscillators analytical solutions
Cveticanin, Livija
2014-01-01
This book provides the presentation of the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. The book presents the original author’s method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter is considered. Special attention is given to the one and two mass oscillatory systems with two-degrees-of-freedom. The criteria for the deterministic chaos in ideal and non-ideal pure nonlinear oscillators are derived analytically. The method for suppressing chaos is developed. Important problems are discussed in didactic exercises. The book is self-consistent and suitable as a textbook for students and also for profess...
Explicit and exact travelling wave solutions for the generalized derivative Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Huang Dingjiang [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)]. E-mail: hdj8116@163.com; Li Desheng [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China); Department of Mathematics, Shenyang Normal University, Shenyang 110034 (China); Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2007-02-15
In this paper, a new auxiliary equation expansion method and its algorithm is proposed by studying a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Being concise and straightforward, the method is applied to the generalized derivative Schroedinger equation. As a result, some new exact travelling wave solutions are obtained which include bright and dark solitary wave solutions, triangular periodic wave solutions and singular solutions. This algorithm can also be applied to other nonlinear wave equations in mathematical physics.
Exact traveling wave solutions to the Klein-Gordon equation using the novel (G‧/G)-expansion method
Hafez, M. G.; Alam, Md. Nur; Akbar, M. Ali
The novel (G‧/G)-expansion method is one of the powerful methods that appeared in recent times for establishing exact traveling wave solutions of nonlinear partial differential equations. Exact traveling wave solutions in terms of hyperbolic, trigonometric and rational functions to the cubic nonlinear Klein-Gordon equation via this method are obtained in this article. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It is shown that the novel (G‧/G)-expansion method is a simple and valuable mathematical tool for solving nonlinear evolution equations (NLEEs) in applied mathematics, mathematical physics and engineering.
Exact traveling wave solutions to the Klein–Gordon equation using the novel (G′/G-expansion method
Directory of Open Access Journals (Sweden)
M.G. Hafez
2014-01-01
Full Text Available The novel (G′/G-expansion method is one of the powerful methods that appeared in recent times for establishing exact traveling wave solutions of nonlinear partial differential equations. Exact traveling wave solutions in terms of hyperbolic, trigonometric and rational functions to the cubic nonlinear Klein–Gordon equation via this method are obtained in this article. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It is shown that the novel (G′/G-expansion method is a simple and valuable mathematical tool for solving nonlinear evolution equations (NLEEs in applied mathematics, mathematical physics and engineering.
Exact Periodic Solitary Solutions to the Shallow Water Wave Equation
Institute of Scientific and Technical Information of China (English)
LI Dong-Long; ZHAO Jun-Xiao
2009-01-01
Exact solutions to the shallow wave equation are studied based on the idea of the extended homoclinic test and bilinear method. Some explicit solutions, such as the one soliton solution, the doubly-periodic wave solution and the periodic solitary wave solutions, are obtained. In addition, the properties of the solutions are investigated.
An exact solution of the slow-light problem
Rybin, A V; Bishop, A R
2004-01-01
We investigate propagation of a slow-light soliton in atomic vapors and Bose-Einstein condensates described by the nonlinear Lambda-model. We show that the group velocity of the soliton monotonically decreases with the intensity of the controlling laser field, which decays exponentially after the laser is switched off. The shock wave of the vanishing controlling field overtakes the slow soliton and stops it, while the optical information is recorded in the medium in the form of spatially localized polarization. We find an explicit exact solution describing the whole process within the slowly varying amplitude and phase approximation. Our results point to the possibility of addressing spatially localized memory formations and moving these memory bits along the medium in a controllable fashion.
New traveling wave solutions for nonlinear evolution equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Madkour, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-06-11
The generalized Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the new exact traveling wave solutions. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in mathematical physics. As a result, many exact traveling wave solutions are obtained which include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.
New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation
Indian Academy of Sciences (India)
Yusuf Pandir
2014-06-01
In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.
An exact approach to intensity analysis of optical pulses in nonlinear meta-materials
Nanda, Lipsa
2016-05-01
The nonlinear pulse propagation has been analytically studied by solving the nonlinear Schrödinger's equation (NLSE) in bulk media exhibiting frequency dependent dielectric permittivity(ɛ) and magnetic permeability(μ). The exact solutions obtained are shown to be of trigonometric & localized types. The analytical and simulation based method has been further extended to investigate the intensity distribution in a nonlinear meta-material which behaves as a negative refractive medium (NRM), where both ɛ and μ are shown to be dispersive and negative in nature.
Integrable KP Coupling and Its Exact Solution
Institute of Scientific and Technical Information of China (English)
彭凌; 杨旭东; 楼森岳
2012-01-01
The integrable coupling is one of the most important topics in the nonlinear physics. This paper creates a novel integrable KP coupling and solves it via a recently-developed dark parameterization procedure.
Exact solution of the robust knapsack problem☆
Monaci, Michele; Pferschy, Ulrich; Serafini, Paolo
2013-01-01
We consider an uncertain variant of the knapsack problem in which the weight of the items is not exactly known in advance, but belongs to a given interval, and an upper bound is imposed on the number of items whose weight differs from the expected one. For this problem, we provide a dynamic programming algorithm and present techniques aimed at reducing its space and time complexities. Finally, we computationally compare the performances of the proposed algorithm with those of different exact algorithms presented so far in the literature for robust optimization problems. PMID:24187428
Exact solutions for a class of quasi-exactly solvable models: A unified treatment
Hatami, N.; Setare, M. R.
2017-07-01
The exact solution of the Schrödinger equation for the four quasi-exactly solvable potentials is presented using the functional Bethe ansatz method. It is shown that all models give rise to the same basic differential equation which is quasi-exactly solvable. The eigenvalues, eigenfunctions and the allowed potential parameters are given for each of the four models in terms of the roots of a set of algebraic Bethe ansatz equations.
An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).
Method of the Logistic Function for Finding Analytical Solutions of Nonlinear Differential Equations
Kudryashov, N. A.
2015-01-01
The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions.logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlev´e test, exact solution
Exact Solutions for the Modified KdV and the Generalized KdV Equations via Exp-Function Method
Directory of Open Access Journals (Sweden)
J. Manafian Heris
2010-06-01
Full Text Available An application of the Exp-function method (EFM to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for the modified KdV equation and the generalized KdV equation. The EFM was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations (NLEEs. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics and applied mathematics.
Exact solutions for steady flows of second-grade fluids
Institute of Scientific and Technical Information of China (English)
ZHANG Dao-xiang; FENG Su-xiao; LU Zhi-ming; LIU Yu-lu
2009-01-01
This paper aims to investigate exact solutions for a second-grade fluid flow with the inverse method.By assuming the relation between the vorticity field and the streamfunction,the exact solutions of the motion of plane second-grade fluids are investigated and obtained.The solutions obtained include simple Couette flows,slit jet flows and uniform flows over a series of distributed obstacles.
Meleshko, Sergey V
2005-01-01
Differential equations, especially nonlinear, present the most effective way for describing complex physical processes. Methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. This book aims to provide scientists, engineers and students with an easy-to-follow, but comprehensive, description of the methods for constructing exact solutions of differential equations.
The Improved Riccati Equation Method and Exact Solutions to mZK Equation
Directory of Open Access Journals (Sweden)
Xiaofeng Li
2012-01-01
Full Text Available We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.
Exact Solutions to Short Pulse Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; ZHENG Ming-Hua; LIU Shi-Kuo
2009-01-01
In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.
Analytic solutions of a class of nonlinearly dynamic systems
Energy Technology Data Exchange (ETDEWEB)
Wang, M-C [System Engineering Institute of Tianjin University, Tianjin, 300072 (China); Zhao, X-S; Liu, X [Tianjin University of Technology and Education, Tianjin, 300222 (China)], E-mail: mchwang123@163.com.cn, E-mail: xszhao@mail.nwpu.edu.cn, E-mail: liuxinhubei@163.com.cn
2008-02-15
In this paper, the homotopy perturbation method (HPM) is applied to solve a coupled system of two nonlinear differential with first-order similar model of Lotka-Volterra and a Bratus equation with a source term. The analytic approximate solutions are derived. Furthermore, the analytic approximate solutions obtained by the HPM with the exact solutions reveals that the present method works efficiently.
DEFF Research Database (Denmark)
Garde, Henrik
2017-01-01
Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity....... For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method...
Construction of exact solutions to the modified forms of DP and CH equations by analytical methods
Directory of Open Access Journals (Sweden)
Jalil Manafian Heris
2015-11-01
Full Text Available In this work, we establish the exact solutions to the modified forms of Degasperis–Procesi (DP and Camassa–Holm (CH equations. The generalized (G’/G-expansion and generalized tanh-coth methods were used to construct solitary wave solutions of nonlinear evolution equations. The generalized (G’/G-expansion method presents a wider applicability for handling nonlinear wave equations. It is shown that the (G’/G-expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
NEW EXPLICIT AND EXACT TRAVELLING WAVE SOLUTIONS FOR A COMPOUND KdV-BURGERS EQUATION
Institute of Scientific and Technical Information of China (English)
XIA TIE-CHENG; ZHANG HONG-QING; YAN ZHEN-YA
2001-01-01
In this paper, new explicit and exact travelling wave solutions for a compound KdV-Burgers equation are obtained by using the hyperbola function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV-Burgers and mKdV equations can be solved by this method. The method can also solve other nonlinear partial differential equations.
Exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation
Indian Academy of Sciences (India)
Yan-Ze Peng
2005-02-01
The extended mapping method with symbolic computation is developed to obtain exact periodic wave solutions to the generalized Nizhnik{Novikov{Veselov equation. Limit cases are studied and new solitary wave solutions and triangular periodic wave solutions are obtained. The method is applicable to a large variety of non-linear partial differential equations, as long as odd- and even-order derivative terms do not coexist in the equation under consideration.
Three dimensional dilatonic gravity's rainbow: exact solutions
Hendi, Seyed Hossein; Panahiyan, Shahram
2016-01-01
Deep relations of dark energy scenario and string theory results with dilaton gravity, on one hand, and the connection between quantum gravity with gravity's rainbow, on the other hand, motivate us to consider three dimensional dilatonic black hole solutions in gravity's rainbow. We obtain two classes of the solutions which are polynomial and logarithmic forms. We also calculate conserved and thermodynamic quantities, and examine the first law of thermodynamics for both classes. In addition, we study thermal stability and show that one of the classes is thermally stable while the other one is unstable.
Searching Exact Solutions for Compact Stars in Braneworld: a conjecture
Ovalle, J
2007-01-01
In the context of the braneworld, a spherically symmetric, static and nonhomogeneous stellar distribution with local and non-local bulk terms is studied. Using a toy solution, it is shown how the general relativistic limit could be lost while a solution is being generated on the brane. The source of this problem is clearly identified and solved by a general solution where a constraint can be identified. This constraint is physically interpreted as a necessary condition to regain general relativity, and a particular solution for it is used to find an exact analytical internal solution to no-uniform stellar distributions on the brane. It is shown that such an exact solution is possible due to the fact that bulk corrections to pressure, density and a metric component are a null source of anisotropic effects on the brane. A conjecture is proposed about the possibility of finding physically relevant exact solutions to non-uniform stellar distributions on the brane.
New exact solutions of the Einstein-Maxwell equations for magnetostatic fields
Institute of Scientific and Technical Information of China (English)
Nisha Goyal; R. K. Gupta
2012-01-01
The symmetry reduction method based on the Fréchet derivative of differential operators is applied to investigate symmetries of the Einstein-Maxwell field equations for magnetostatic fields,which is a coupled system of nonlinear partial differential equations of the second order.The technique yields invariant transformations that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations.Some of the reduced systems are further studied to obtain the exact solutions.
Exact solutions of some fractional differential equations by various expansion methods
Topsakal, Muammer; Guner, Ozkan; Bekir, Ahmet; Unsal, Omer
2016-10-01
In this paper, we construct the exact solutions of some nonlinear spacetime fractional differential equations involving modified Riemann-Liouville derivative in mathematical physics and applied mathematics; namely the fractional modified Benjamin-Bona- Mahony (mBBM) and Kawahara equations by using G'/G and (G'/G, 1/G)-expansion methods.
Peakons and new exact solitary wave solutions of extended quantum Zakharov-Kuznetsov equation
Zhang, Ben-gong; Li, Weibo; Li, Xiangpeng
2017-06-01
In this paper, the three dimensional extended quantum Zakharov-Kuznetsov equation, which arises in the dimensionless hydrodynamic equations describing the nonlinear propagation of the quantum ion-acoustic waves, is investigated by an auxiliary equation method. As a result, peakons and a series of new exact traveling wave solutions, including bell-shaped, kink-type solitary wave, shock wave, periodic wave, and Jacobi elliptic solutions, are obtained. We also analyze the three kinds of nonlinear structures of our results, i.e., blowup, peakons, and shock wave. These new exact solutions will enrich the previous results and help us to further understand the physical structures and analyze the nonlinear propagation of the quantum ion-acoustic waves.
On exact solutions in multidimensional gravity with antisymmetric forms
Ivashchuk, V D
2001-01-01
This short review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the product form. The sigma-model approach and exact solutions are reviewed.
New Exact Solutions to Long-Short Wave Interaction Equations
Institute of Scientific and Technical Information of China (English)
TIAN Ying-Hui; CHEN Han-Lin; LIU Xi-Qiang
2006-01-01
New exact solutions expressed by the Jacobi elliptic functions are obtained to the long-short wave interaction equations by using the modified F-expansion method. In the limit case, solitary wave solutions and triangular periodic wave solutions are obtained as well.
Exact solution of the neutron transport equation in spherical geometry
Energy Technology Data Exchange (ETDEWEB)
Anli, Fikret; Akkurt, Abdullah; Yildirim, Hueseyin; Ates, Kemal [Kahramanmaras Suetcue Imam Univ. (Turkey). Faculty of Sciences and Letters
2017-03-15
Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using P{sub N} approximation which is widely used in neutron transport theory.
Some exact solutions in K-essence theory isotropic cosmology
Pimentel, Luis O
2016-01-01
We use a simple form of the K-essence theory and apply it to the classic isotropic cosmological model and seek exact solutions. The particular form of the kinetic term that we choose is $K \\left(\\phi, X \\right)= K_0(\\phi)X^m +K_1$. The resulting field equations in the homogeneous and isotropic cosmology (FRW)is considered. Several exact solutions are obtained.
Energy Technology Data Exchange (ETDEWEB)
Zhang Huiqun [College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071 (China)], E-mail: hellozhq@yahoo.com.cn
2009-02-15
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
Nonequivalent Similarity Reductions and Exact Solutions for Coupled Burgers-Type Equations
M. H. M., Moussa; R. A. K., Omar; Rehab, M. El-Shiekh; H. R., El-Melegy
2012-01-01
Using the machinery of Lie group analysis, the nonlinear system of coupled Burgers-type equations is studied. Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras, it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations. The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions, hyperbolic functions, and trigonometric functions. Some figures are given to show the properties of the solutions.
Nonequivalent Similarity Reductions and Exact Solutions for Coupled Burgers-Type Equations
Institute of Scientific and Technical Information of China (English)
M.H.M. Moussa; R.A.K. Omar; Rehab M. El-Shiekh; H.R. El-Melegy
2012-01-01
Using the machinery of Lie group analysis, the nonlinear system of coupled Burgers-type equations is studied. Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras, it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations. The search for solutions of these systems by using the G/G-method has yielded certain exact solutions expressed by rational functions, hyperbolic functions, and trigonometric functions. Some figures are given to show the properties of the solutions.
New Exact Solutions for Isothermal Magnetostatic Atmosphere Equations
Directory of Open Access Journals (Sweden)
Mohamed Tawfik ATTIA
2014-12-01
Full Text Available Here, an extended, (G'/G-expansion method with a computerized symbolic computation is used for constructing the exact travelling wave solutions for isothermal magnetostatic atmospheres equations. These equations depend on arbitrary functions that must be specified with choices of the different choice of the different arbitrary functions. The proposed method has been successfully used to obtain some exact travelling wave solutions for the Liouville and sinh-Poisson equations. The obtained travelling wave solutions are expressed by hyperbolic, triangular and exponential function. The solutions obtained via the propose method have many potential applications in physics.
Anisotropic exact solutions in scalar-tensor-vector gravity
Sharif, M.; Yousaf, Aasma
2016-09-01
The aim of this paper is to explore exact solutions in the scalar-tensor-vector theory of gravity with two scalar fields and one vector field. We consider a locally rotationally symmetric Bianchi type-I universe filled with perfect fluid. The first exact solution is found through certain assumptions while the second solution is obtained through Noether symmetry approach. We discuss the behavior of the resulting solutions numerically and also explore the corresponding energy conditions. It is found that the strong energy condition is violated in both cases indicating the accelerated expansion of the universe.
The First-Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations
Directory of Open Access Journals (Sweden)
Yadong Shang
2012-01-01
Full Text Available This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.
Instantaneous Bethe-Salpeter Equation and Its Exact Solution
Institute of Scientific and Technical Information of China (English)
CHANG Chao-Hsi; CHEN Jiao-Kai; LI Xue-Qian; WANG Guo-Li
2005-01-01
We present an approach to solve Bethe-Salpeter (BS) equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and welldetermined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium,the exact solutions precisely present corrections to those of the corresponding Schrodinger equation in order v1 (v is the relative velocity) for eigenfunctions, in order v2 for eigenvalues, and the mixing between S and D components in JPC = 1- states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the O(v) corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states.
Fully nonlinear and exact perturbations of the Friedmann world model
Hwang, Jai-chan
2012-01-01
In 1988 Bardeen has suggested a pragmatic formulation of cosmological perturbation theory which is powerful in practice to employ various fundamental gauge conditions easily depending on the character of the problem. The perturbation equations are presented without fixing the temporal gauge condition and are arranged so that one can easily impose fundamental gauge conditions by simply setting one of the perturbation variables in the equations equal to zero. In this way one can use the gauge degrees of freedom as an advantage in handling problems. Except for the synchronous gauge condition, all the other fundamental gauge conditions completely fix the gauge mode, and consequently, each variable in such a gauge has a unique gauge invariant counterpart, so that we can identify the variable as the gauge-invariant one. Here, we extend Bardeen's linear formulation to fully nonlinear order in perturbations, with the gauge advantage kept intact. Derived equations are exact, and from these we can easily expand to high...
Exact solutions for logistic reaction-diffusion equations in biology
Broadbridge, P.; Bradshaw-Hajek, B. H.
2016-08-01
Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.
Exact solutions to the supply chain equations for arbitrary, time-dependent demands
DEFF Research Database (Denmark)
Warburton, Roger D.H.; Hodgson, J.P.E.; Nielsen, Erland Hejn
2014-01-01
We study the impact on inventory of an unexpected, non-linear, time-dependent demand and present the exact solutions over time to the supply chain equations without requiring any approximations. We begin by imposing a boundary condition of stability at infinity, from which we derive expressions...... for the estimated demand and the target work in progress when the demand is time-dependent. The resulting inventory equation is solved in terms of the Lambert modes with all of the demand non-linearities confined to the pre-shape function. The series solution is exact, and all terms are reasonably easy to calculate......, so users can determine the inventory behavior to any desired precision. To illustrate, we solve the equations for a non-linear, quadratic time-dependence in the demand. For practical use, only a few terms in the series are required, a proposition illustrated by the For All Practical Purposes (FAPP...
Exact solutions to the supply chain equations for arbitrary, time-dependent demands
DEFF Research Database (Denmark)
Warburton, Roger D.H.; Hodgson, J.P.E.; Nielsen, Erland Hejn
2014-01-01
, so users can determine the inventory behavior to any desired precision. To illustrate, we solve the equations for a non-linear, quadratic time-dependence in the demand. For practical use, only a few terms in the series are required, a proposition illustrated by the For All Practical Purposes (FAPP......We study the impact on inventory of an unexpected, non-linear, time-dependent demand and present the exact solutions over time to the supply chain equations without requiring any approximations. We begin by imposing a boundary condition of stability at infinity, from which we derive expressions...... for the estimated demand and the target work in progress when the demand is time-dependent. The resulting inventory equation is solved in terms of the Lambert modes with all of the demand non-linearities confined to the pre-shape function. The series solution is exact, and all terms are reasonably easy to calculate...
Exact Analytical Solutions in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length
Institute of Scientific and Technical Information of China (English)
CHEN Yong; LI Biao; ZHENG Yu
2007-01-01
In the paper, the generalized Riccati equation rational expansion method is presented. Making use of the method and symbolic computation, we present three families of exact analytical solutions of Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Then the dynamics of two anlytical solutions are demonstrated by computer simulations under some selectable parameters including the Feshbach-managed nonlinear coefficient and the hyperbolic secant function coefficient.
New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin; GUO Jun; ZHAO Hong
2005-01-01
In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like,period-form solutions of nonlinear evolution equations (NEEs). This method is more powerful than the extended tanhfunction method [Phys. Lett. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Lett. A 285 (2001) 355]. Abundant new families of the exact solutions of Bogoyavlenskii's generalized breaking soliton equation are obtained by using this method and symbolic computation system Maple.
New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation
Institute of Scientific and Technical Information of China (English)
BAICheng-Lin; GUOJun; ZHAOHong
2005-01-01
In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like,period-form solutions of nonlinear evolution equations (NEEs). This method is more powerful than the extended tanhfunction method [Phys. Left. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Left. A 285 (2001) 355]. Abundant new families of the exact solutions of Bogoyavlenskii's generalized breaking soliton equation are obtained by using this method and symbolic computation system Maple.
The Exact Traveling Wave Solutions to Two Integrable KdV6 Equations
Institute of Scientific and Technical Information of China (English)
Jibin LI; Yi ZHANG
2012-01-01
The exact explicit traveling solutions to the two completely integrable sixthorder nonlinear equations KdV6 are given by using the method of dynamical systems and Cosgrove's work.It is proved that these traveling wave solutions correspond to some orbits in the 4-dimensional phase space of two 4-dimensional dynamical systems.These orbits lie in the intersection of two level sets defined by two first integrals.
General Exact Solutions of the Harry—Dym Equation
Reza, Mokhtari
2011-02-01
The aim of this paper is to generate exact travelling wave solutions of the Harry—Dym equation through the methods of Adomian decomposition, He's variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.
AdS Waves as Exact Solutions to Quadratic Gravity
Gullu, Ibrahim; Sisman, Tahsin Cagri; Tekin, Bayram
2011-01-01
We give an exact solution of the quadratic gravity in D dimensions. The solution is a plane fronted wave metric with a cosmological constant. This metric solves not only the full quadratic gravity field equations but also the linearized ones which include the linearized equations of the recently found critical gravity.
Exact solutions, energy, and charge of stable Q-balls
Energy Technology Data Exchange (ETDEWEB)
Bazeia, D.; Marques, M.A. [Universidade Federal da Paraiba, Departamento de Fisica, Joao Pessoa, PB (Brazil); Menezes, R. [Universidade Federal da Paraiba, Departamento de Ciencias Exatas, Rio Tinto, PB (Brazil); Universidade Federal de Campina Grande, Departamento de Fisica, Campina Grande, PB (Brazil)
2016-05-15
In this work we deal with nontopological solutions of the Q-ball type in two spacetime dimensions. We study models of current interest, described by a Higgs-like and other, similar potentials which unveil the presence of exact solutions. We use the analytic results to investigate how to control the energy and charge to make the Q-balls stable. (orig.)
Exact angular momentum projection based on cranked HFB solution
Energy Technology Data Exchange (ETDEWEB)
Enami, Kenichi; Tanabe, Kosai; Yosinaga, Naotaka [Saitama Univ., Urawa (Japan). Dept. of Physics
1998-03-01
Exact angular momentum projection of cranked HFB solutions is carried out. It is reconfirmed from this calculation that cranked HFB solutions reproduce the intrinsic structure of deformed nucleus. The result also indicates that the energy correction from projection is important for further investigation of nuclear structure. (author)
Exact solutions for classical Yang-Mills fields
Frasca, Marco
2014-01-01
Some years ago we displayed a set of classical solutions for the classical Yang-Mills field theory having the property to satisfy a dispersion relation typical of a massive theory. But such solutions seemed to be exact only in the Landau gauge making all the argument an asymptotic one for the most general case of a generic gauge. These solutions can be used to describe the vacuum of the quantum Yang-Mills theory and so, to prove that they are always exact can grant a general framework to build a quantum field theory. Here we show that these solutions are always exact changing just the normalization factor. The components of the field become separated on a generic gauge being all equal just in the Landau gauge.
New Exact Solutions of Boussinesq Equation%Boussinesq 方程新的精确解
Institute of Scientific and Technical Information of China (English)
杨琼芬; 杜先云; 杨立娟
2011-01-01
以齐次平衡原则和试探函数法为基础,给出函数变换与双线性算子相结合的方法,构造了Boussinesq方程新的精确解.%Based on the homogeneous balance principle and the trial function method, a method for combining function transformation with bilinear operator is proposed. . And the method is applied to construct new exact solutions of Boussinesq equation. This method can be used to find new exact solutions to other nonlinear evolution equations.
Directory of Open Access Journals (Sweden)
Mehmet Tarik Atay
2013-01-01
Full Text Available The Variational Iteration Method (VIM and Modified Variational Iteration Method (MVIM are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational Iteration Method (VIM and the Modified Variational Iteration Method (MVIM are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.
A family of exact solutions for unpolarized Gowdy models
Obregón, O; Obregon, Octavio; Ryan, Michael P.
1998-01-01
Unpolarized Gowdy models are inhomogeneous cosmological models that depend on time and one spatial variable and have complicated nonlinear equations of motion. There are two topologies associated with these models, a three-torus and a one-sphere cross a two-sphere. The three-torus models have been used for numerical studies because it seems difficult to find analytic solutions to their nonlinear Einstein equations. The one-sphere cross tow-sphere models have even more complicated equations, but at least one family of analytic solutions can be given as a reinterpretation of known solutions. Various properties of this family of solutions are studied.
Relativit\\'e g\\'en\\'erale: solutions exactes stationnaires
Clément, Gérard
2011-01-01
After a brief summary of the foundations of general relativity, we will concentrate on the stationary exact solutions of the Einstein and Einstein-Maxwell equations. A number of these solutions can be interpreted as black holes, corresponding to a special spacetime geometry. Then, we will show that the reformulation of the stationary Einstein-Maxwell equations as a gravitating sigma model provides us with a powerful tool to generate new solutions.
Exact interior solutions in 2 + 1-dimensional spacetime
Energy Technology Data Exchange (ETDEWEB)
Rahaman, Farook; Bhar, Piyali [Jadavpur University, Department of Mathematics, Kolkata, West Bengal (India); Biswas, Ritabrata [Indian Institute of Engineering Sceince and Technology Shibpur, Howrah, West Bengal (India); Usmani, A.A. [Aligarh Muslim University, Department of Physics, Aligarh, Uttar Pradesh (India)
2014-04-15
We provide a new class of exact solutions for the interior in 2 + 1-dimensional spacetime. The solutions obtained for the perfect fluid model both with and without cosmological constant (Λ) are found to be regular and singularity free. It assumes very simple analytical forms that help us to study the various physical properties of the configuration. Solutions without Λ are found to be physically acceptable. (orig.)
Exact solutions for the family of third order Korteweg de-Vries equations
2016-01-01
In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV equations. The solutions are expressed in terms of hyperbolic functions, trigonometric functions and...
Exact solution of the space-time fractional coupled EW and coupled MEW equations
Raslan, K. R.; S. EL-Danaf, Talaat; K. Ali, Khalid
2017-07-01
In this paper, we obtained a traveling wave solution by using the Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations, such as the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEWE), which are the important soliton equations. Both equations are reduced to ordinary differential equations by use of the fractional complex transform and of the properties of the modified Riemann-Liouville derivative. We plot the exact solutions for these equations at different time levels.
New Exact Solutions to NLS Equation and Coupled NLS Equations
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2004-01-01
A transformation is introduced on the basis of the projective Riccati equations, and it is applied as an intermediate in expansion method to solve nonlinear Schrodinger (NLS) equation and coupled NLS equations. Many kinds of envelope travelling wave solutions including envelope solitary wave solution are obtained, in which some are found for the first time.
Searching Exact Solutions for Compact Stars in Braneworld:. a Conjecture
Ovalle, J.
In the context of the braneworld, a method to find consistent solutions to Einstein's field equations in the interior of a spherically symmetric, static and non-uniform stellar distribution with Weyl stresses is developed. This method, based on the fact that any braneworld stellar solution must have the general relativity solution as a limit, produces a constraint which reduces the degrees of freedom on the brane. Hence the nonlocality and non-closure of the braneworld equations can be overcome. The constraint found is physically interpreted as a necessary condition to regain general relativity, and a particular solution for it is used to find an exact and physically acceptable analytical internal solution to no-uniform stellar distributions on the brane. It is shown that such an exact solution is possible due to the fact that bulk corrections to pressure, density and a metric component are a null source of anisotropic effects on the brane. A conjecture is proposed regarding the possibility of finding physically relevant exact solutions to non-uniform stellar distributions on the brane.
New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods
S Saha, Ray
2016-04-01
In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.
Analyzing Multi-Field Tunneling With Exact Bounce Solutions
Aravind, Aditya; Lorshbough, Dustin; Paban, Sonia
2014-01-01
We study multi-field tunneling using exact solutions for additive potentials. We introduce a binomial potential with non-integer powers that could be considered a generalization of the $4D$ Fubini instanton potential. Using scaling arguments, we show that for multi-field potentials taller and wider barriers may still lead to a smaller bounce action.
Feynman graph solution to Wilson's exact renormalization group
Sonoda, H
2003-01-01
We introduce a new prescription for renormalizing Feynman diagrams. The prescription is similar to BPHZ, but it is mass independent, and works in the massless limit as the MS scheme with dimensional regularization. The prescription gives a diagrammatic solution to Wilson's exact renormalization group differential equation.
Exact Solutions of the Photon Equation in some Rotating Universes
Aydogdu, O; Korunur, M; Salti, M
2005-01-01
In this paper exact solutions of the photon equation are presented and the second order differential equation form of the Maxwell equations in some rotating universes is found to show the equivalence with the Photon equation which is obtained the same form for the same universes.
Gravitational waves as exact solutions of Einstein field equations
Energy Technology Data Exchange (ETDEWEB)
Vilasi, G [Dipartimento di Fisica, Universita di Salerno Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Gruppo Collegato di Salerno Via S. Allende, I-84081 Baronissi (Salerno) (Italy)
2007-11-15
Exact solutions of Einstein field equations invariant for a non-Abelian 2-dimensional Lie algebra of Killing fields are described. A sub-class of these gravitational fields have a wave-like character; it is shown that they have spin-1.
Exact solution to surface displacement associated with sources ...
African Journals Online (AJOL)
user
International Journal of Engineering, Science and Technology ... Usually an exact solution to the surface displacement in an elastic half space is available for ..... to (ζ1 , ζ 2 ,ζ3 ) by means of the transformation (2.1), i.e, the station (x, y,-h) on.
Exact solutions for classical Yang-Mills fields
2014-01-01
We provide a set of exact solutions of the classical Yang-Mills equations. They have the property to satisfy a massive dispersion relation and hold in all gauges. These solutions can be used to describe the vacuum of the quantum Yang-Mills theory and so, they provide a general framework to build a quantum field theory. The components of the field become separated on a generic gauge but are all equal just in the Lorenz (Landau) gauge.
Nonlocal Symmetries and Exact Solutions for PIB Equation
Xin, Xiang-Peng; Miao, Qian; Chen, Yong
2012-09-01
In this paper, the symmetry group of the (2+1)-dimensional Painlevé integrable Burgers (PIB) equations is studied by means of the classical symmetry method. Ignoring the discussion of the infinite-dimensional subalgebra, we construct an optimal system of one-dimensional group invariant solutions. Furthermore, by using the conservation laws of the reduced equations, we obtain nonlocal symmetries and exact solutions of the PIB equations.
Nonlocal Symmetries and Exact Solutions for PIB Equation
Institute of Scientific and Technical Information of China (English)
辛祥鹏; 苗倩; 陈勇
2012-01-01
In this paper, the symmetry group of the is studied by means of the classical symmetry method （2＋l）-dimensionM Painlevd integrable Burgers （PIB） equations Ignoring the discussion of the infinite-dimensional subalgebra, we construct an optimal system of one-dimensional group invariant solutions. Furthermore, by using the conservation laws of the reduced equations, we obtain nonlocal symmetries and exact solutions of the PIB equations.
The exact solution of an octagonal rectangle triangle random tiling
De Gier, J; Gier, Jan de; Nienhuis, Bernard
1996-01-01
We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eight-fold symmetric phase. The solution is obtained using Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling, which were found by M. Widom.
Stable and Critical Noncohesive Coulomb Wedges: Exact Elastic Solutions
Wang, K.; Hu, Y.
2004-12-01
The theory of critically tapered Coulomb wedge has been successfully applied to model active fold-and-thrust belts or submarine accretionary prisms. Brittle mountain building is episodic in nature, controlled by changes in basal friction, erosion and sedimentation, and hydrogeology. Sediment accretion may be modulated by great subduction earthquakes. Between deformation episodes and/or during transition between compressional and extensional tectonics, the Coulomb wedges are stable (i.e., supercritical), to which the critical taper theory does not apply. In this work, we provide an exact elastic solution for stable wedges based on Airy stress functions. The stress equilibrium equation and definition of basal friction and basal and internal pore fluid pressure ratios are exactly the same as those used for Dahlen's [1984] exact solution for critical noncohesive Coulomb wedges, but internal friction μ becomes irrelevant. Given elastic - perfectly Coulomb-plastic rheology, for stresses in a wedge on the verge of Coulomb failure there must co-exist a critical taper solution involving μ and a unique equivalent elastic solution not involving μ . Our elastic solution precisely reduces to Dahlen's critical taper solution for critical conditions. For stable conditions, normal stress perpendicular to the surface slope σ z and shear stress τ xz are identical with those in a critical taper, but the slope-parallel normal stress is different. The elastic solution is also generally applicable to purely elastic wedges and useful for modeling geodetic observations. A stable noncohesive Coulomb wedge differs from a general elastic wedge in that its upper and lower surfaces stay at zero curvature during loading. Dahlen, F.A. (1984), Noncohesive critical Coulomb wedges: An exact solution, JGR, 89, 10,125-10,133.
Exact Penalty Function and Asymptotic Strong Nonlinear Duality in Integer Programming
Institute of Scientific and Technical Information of China (English)
Fu-sheng Bai; Z.Y.Wu; L.S. Zhang
2004-01-01
In this paper, a logarithmic-exponential penalty function with two parameters for integer programmingis discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties.The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].
A New Kind of Simple Smooth Exact Penalty Function of Constrained Nonlinear Programming
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The penalty function method is one basic method for solving constrained nonlinear programming, in which simple smooth exact penalty functions draw much attention for their simpleness and smoothness. This article offers a new kind of simple smooth approximative exact penalty function of general constrained nonlinear programmings and analyzes its properties.
A Direct Algebraic Method in Finding Particular Solutions to Some Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
LIUChun-Ping; CHENJian-Kang; CAIFan
2004-01-01
Firstly, a direct algebraic method and a routine way in finding traveling wave solutions to nonlinear evolution equations are explained. And then some new exact solutions for some evolution equations are obtained by using the method.
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Directory of Open Access Journals (Sweden)
Wei Li
2014-01-01
Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
Quantifying risks with exact analytical solutions of derivative pricing distribution
Zhang, Kun; Liu, Jing; Wang, Erkang; Wang, Jin
2017-04-01
Derivative (i.e. option) pricing is essential for modern financial instrumentations. Despite of the previous efforts, the exact analytical forms of the derivative pricing distributions are still challenging to obtain. In this study, we established a quantitative framework using path integrals to obtain the exact analytical solutions of the statistical distribution for bond and bond option pricing for the Vasicek model. We discuss the importance of statistical fluctuations away from the expected option pricing characterized by the distribution tail and their associations to value at risk (VaR). The framework established here is general and can be applied to other financial derivatives for quantifying the underlying statistical distributions.
Algebraic dynamics solution to and algebraic dynamics algorithm for nonlinear advection equation
Institute of Scientific and Technical Information of China (English)
2008-01-01
Algebraic dynamics approach and algebraic dynamics algorithm for the solution of nonlinear partial differential equations are applied to the nonlinear advection equa-tion. The results show that the approach is effective for the exact analytical solu-tion and the algorithm has higher precision than other existing algorithms in nu-merical computation for the nonlinear advection equation.
Directory of Open Access Journals (Sweden)
Isaiah Elvis Mhlanga
2012-01-01
Full Text Available We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1-dimensional Davey-Stewartson equations.
Exact Solutions for Models of Cultural Transmission and Network Rewiring
Evans, T S
2006-01-01
We look at the evolution through rewiring of the degree distribution of a network so the number edges is constant. This is exactly equivalent to the evolution of probability distributions in models of cultural transmission with drift and innovation, or models of homogeneity in genes in the presence of mutation. We show that the mean field equations in the literature are incomplete and provide the full equations. We then give an exact solution for both their long time solution and for their approach to equilibrium. Numerical results show these are excellent approximations and confirm the characteristic simple inverse power law distributions with a large scale cutoff under certain conditions. The alternative is that we reach a completely homogeneous solution. We consider how such processes may arise in practice, using a recent Minority Game study as an example.
Exact solutions and zero modes in scalar field theory
Frasca, Marco
2013-01-01
We provide a set of exact solutions in field theory of scalar fields with $Z_2$ symmetry that involve Jacobi elliptic functions. These solutions have the interesting property to provide massive waves even if one starts from a massless equation. We analyze them classically providing also exact solutions to the corresponding equations for the Green functions needed to completely solve them in a strong coupling limit. This is accomplished using a functional expansion into powers of the current. It is shown that the spontaneous breaking of the $Z_2$ symmetry is due to the existence of a zero mode that persists also in the case of the Higgs mechanism. In this latter case, the zero mode seems to play a role similar to the Goldstone boson in the breaking of a continuous symmetry and so, it should be important at lower momenta as a long range excitation.
New Exact Solutions of Ion-Acoustic Wave Equations by (G′/G-Expansion Method
Directory of Open Access Journals (Sweden)
Wafaa M. Taha
2013-01-01
Full Text Available The (G′/G-expansion method is used to study ion-acoustic waves equations in plasma physic for the first time. Many new exact traveling wave solutions of the Schamel equation, Schamel-KdV (S-KdV, and the two-dimensional modified KP (Kadomtsev-Petviashvili equation with square root nonlinearity are constructed. The traveling wave solutions obtained via this method are expressed by hyperbolic functions, the trigonometric functions, and the rational functions. In addition to solitary waves solutions, a variety of special solutions like kink shaped, antikink shaped, and bell type solitary solutions are obtained when the choice of parameters is taken at special values. Two- and three-dimensional plots are drawn to illustrate the nature of solutions. Moreover, the solution obtained via this method is in good agreement with previously obtained solutions of other researchers.
Institute of Scientific and Technical Information of China (English)
李凯辉; 刘汉泽; 辛祥鹏
2016-01-01
The symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a signif-icant role in nonlinear science and mathematical physics. Symmetry is derived from physics, and it is a mathematical description for invariance. Symmetry group theory plays an important role in constructing explicit solutions, whether the equations are integrable or not. By using the symmetry method, an original nonlinear system can be reduced to a system with fewer independent variables through any given subgroup. But, since there are almost always an infinite number of such subgroups, it is usually not feasible to list all possible group invariant solutions to the system. It is anticipated to find all those equivalent group invariant solutions, that is to say, to construct the one-dimensional optimal system for the Lie algebra. Construction of explicit forms of conservation laws is meaningful, as they are used for developing the appropriate numerical methods and for making mathematical analyses, in particular, of existence, uniqueness and stability. In addition, the existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability. The similarity solutions are of importance for investigating the long-time behavior, blow-up profile and asymptotic phenomena of a non-linear system. For instance, in some circumstance, the asymptotic behaviors of finite-mass solutions of non-linear diffusion equation with non-linear source term are described by an explicit self-similar solution, etc. However, how to tackle these matters is a complicated problem that challenges researchers to be solved. In this paper, by using the symmetry method, we obtain the symmetry reduction, optimal systems, and many new exact group invariant solution of a fifth-order nonlinear wave equation. By Lie symmetry analysis method, the point symmetries and an optimal system of the equation are obtained. The exact power
A static axisymmetric exact solution of f(R)-gravity
Energy Technology Data Exchange (ETDEWEB)
Gutierrez-Pineres, Antonio C., E-mail: acgutierrez@correo.nucleares.unam.mx [Facultad de Ciencias Basicas, Universidad Tecnologica de Bolivar, CO 131001 Cartagena de Indias (Colombia); Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, A.P. 70-543, 04510 Mexico D.F. (Mexico); Lopez-Monsalvo, Cesar S., E-mail: cesar.slm@correo.nucleares.unam.mx [Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, A.P. 70-543, 04510 Mexico D.F. (Mexico)
2013-01-29
We present an exact, axially symmetric, static, vacuum solution for f(R)-gravity in Weyl's canonical coordinates. We obtain a general explicit expression for the dependence of df(R)/dR upon the r and z coordinates and then the corresponding explicit form of f(R), which must be consistent with the field equations. We analyze in detail the modified Schwarzschild solution in prolate spheroidal coordinates. Finally, we study the curvature invariants and show that, in the case of f(R){ne}R, this solution corresponds to a naked singularity.
EXACT AUGMENTED LAGRANGIAN FUNCTION FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS
Institute of Scientific and Technical Information of China (English)
DU Xue-wu; ZHANG Lian-sheng; SHANG You-lin; LI Ming-ming
2005-01-01
An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions,the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, from the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.
Directory of Open Access Journals (Sweden)
Ma Hong-Cai
2013-01-01
Full Text Available The simple direct method is adopted to find Non-Auto-Backlund transformation for variable coefficient non-linear systems. The (2+1-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients is used as an example to elucidate the solution procedure, and its symmetry transformation and exact solutions are obtained.
Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2012-01-01
Full Text Available We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.
Exact solutions for the family of third order Korteweg de-Vries equations
Directory of Open Access Journals (Sweden)
Luwai Wazzan
2016-05-01
Full Text Available In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV equations, namely, the KdV equation, the modified KdV (mKdV equation, the potential KdV (pKdV equation, the generalized KdV (gKdV equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV equations. The solutions are expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The method used is promising method to solve other nonlinear evaluation equations.
Exact, Schwarzschild-like solution for Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Singleton, D.
1995-04-01
Exploiting the connection between general relativity and Yang-Mills theory an exact, Schwarzchild-like solution is given for an SU(N) gauge field coupled to a scalar field in the Bogomolny, Prasad, Sommerfield limit. The SU(2) solution is found using the second order Euler-Lagrange formalism, while the SU(N) generalization is given using the first order Bogomolny formalism. In analogy with the Schwarzschild solution of general relativity, these Yang-Mills solutions possess an event horizon with respect to the SU(N) charge. It is conjectured that this may be the confinement mechanism for QCD, since just as a Schwarzschild black hole will permanently confine anything which carries the charge of general relativity (mass-energy), so this Yang-Mills solution will confine any particle which carries the SU(N) charge.
A new method of generating exact inflationary solutions
Schunck, F E; Franz E Schunck; Eckehard W Mielke
1994-01-01
The mechanism of the initial inflation of the universe is based on gravitationally coupled scalar fields \\phi. Various scenarios are distinguished by the choice of an {\\it effective self--interaction potential} U(\\phi) which simulates a {\\it temporarily} non--vanishing {\\em cosmological term}. Using the Hubble expansion parameter H as a new ``time" coordinate, we can formally derive the {\\it general} Robertson--Walker metric for a {\\em spatially flat} cosmos. Our new method provides a classification of allowed inflationary potentials and is broad enough to embody all known {\\it exact} solutions involving one scalar field as special cases. Moreover, we present new inflationary and deflationary exact solutions and can easily predict the influence of the form of U(\\phi) on density perturbations.
On the Exact Solution of Wave Equations on Cantor Sets
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2015-09-01
Full Text Available The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM. We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs. The efficiency of the scheme is examined by two illustrative examples.
Some exact BPS solutions for exotic vortices and monopoles
Ramadhan, Handhika S.
2016-07-01
We present several analytical solutions of BPS vortices and monopoles in the generalized Abelian Maxwell-Higgs and Yang-Mills-Higgs theories, respectively. These models have recently been extensively studied and several exact solutions have already been obtained in [1,2]. In each theory, the dynamics is controlled by the additional two positive scalar-field-dependent functions, f (| ϕ |) and w (| ϕ |). For the case of vortices, we work in the ordinary symmetry-breaking Higgs potential, while for the case of monopoles we have the ordinary condition of the Prasad-Sommerfield limit. Our results generalize the exact solutions found previously. We also present solutions for BPS vortices with higher winding number. These solutions suffer from the condition that w (| ϕ |) has negative value at some finite range of r, but we argue that since it satisfies the weaker positive-value conditions then the corresponding energy density is still positive-definite and, thus, they are acceptable BPS solutions.
Symmetries and Exact Solutions of the Breaking Soliton Equation
Institute of Scientific and Technical Information of China (English)
陈美; 刘希强
2011-01-01
With the aid of the classical Lie group method and nonclassical Lie group method, we derive the classical Lie point symmetry and the nonclassical Lie point symmetry of （2＋1）-dimensional breaking soliton （BS） equation. Using the symmetries, we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation. Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.
Exact solutions to quadratic gravity generated by a conformal method
Pravda, Vojtech; Podolsky, Jiri; Svarc, Robert
2016-01-01
We study the role of conformal transformations in constructing vacuum solutions to quadratic gravity. We find that such solutions can be obtained by solving one non-linear partial differential equation for the conformal factor on any Einstein spacetime or, more generally, on any background with vanishing Bach tensor. We show that all spacetimes conformal to Kundt are either Kundt or Robinson--Trautmann, and we provide explicit Kundt and Robinson--Trautman solutions to quadratic gravity by solving the above mentioned equation on certain Kundt backgrounds.
Exact solutions in multidimensional gravity with antisymmetric forms
Ivashchuk, V D
2001-01-01
This topical review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the form M = M_0 x M_1 x ...x M_n, where M_i are Einstein spaces (i >0). The sigma-model approach and exact solutions in the model are reviewed and the solutions with p-branes (e.g. Majumdar-Papapetrou-type, cosmological, spherically symmetric, black-brane and Freund-Rubin-type ones) are considered.
Exact Solution in the New Inflation Scenario with Induced Gravity
Institute of Scientific and Technical Information of China (English)
WANG Wen-Fu
2001-01-01
Taking the Hubble parameter directly as a function of the scalar field instead of as a function of time,H = H( ), we present a new exact solution in the new inflation model with induced gravity. This includes solution which is inflation for end, and develops smoothly towards radiation-like evolution for ≥ end. The inflation is driven by the evolution of the field with inflation potential, V( ) = λ 2 v2)2.density, ns, is computed and ns lies well inside the limits set by the cosmic background explorer (COBE) satellite.the dex of the scalar effective cosmological constant Aeff tends to zero when inflation ends.``
Conformal Invariance and the exact solution of BFKL equations
Navelet, H
1997-01-01
The conformal invariance properties of the QCD Pomeron in the transverse plane allow us to give an explicit analytical expression for the solution of the BFKL equations both in the transverse coordinate and momentum spaces. This result is obtained from the solution of the conformal eigenvectors in the mixed representation in terms of two conformal blocks, each block being the product of an holomorphic times an antiholomorphic function. This property is used to give an exact expression for the QCD dipole multiplicities and dipole-dipole cross-sections in the whole parameter space, proving the equivalence between the BFKL and dipole representations of the QCD Pomeron.
Scalar triplet on a domain wall: an exact solution
Gani, Vakhid A; Radomskiy, Roman V
2016-01-01
We study a model with a real scalar Higgs field and a scalar triplet field that allows existence of a topological defect -- a domain wall. The wall breaks the global $O(3)$ symmetry of the model, which gives rise to non-Abelian orientational degrees of freedom. We found an exact analytic solution that describes a domain wall with a localized configuration of the triplet field on it. This solution enables one to calculate contributions to the action from the orientational and translational degrees of freedom of the triplet field. We also study the linear stability of the domain wall with the triplet field switched off.
On the Exact Solution of a Generalized Polya Process
Directory of Open Access Journals (Sweden)
Hidetoshi Konno
2010-01-01
Full Text Available There are two types of master equations in describing nonequilibrium phenomena with memory effect: (i the memory function type and (ii the nonstationary type. A generalized Polya process is studied within the framework of a non-stationary type master equation approach. For a transition-rate with an arbitrary time-dependent relaxation function, the exact solution of a generalized Polya process is obtained. The characteristic features of temporal variation of the solution are displayed for some typical time-dependent relaxation functions reflecting memory in the systems.
Exact Solution in Chaotic Inflation Model with Negative Potential
Institute of Scientific and Technical Information of China (English)
王文福
2003-01-01
New exact solution to the Einstein equations that describe the evolution of cosmological chaotic inflation model is derived. The inflation is driven by the evolution of scalar field with negative potential V( )=-Vo +1/2m2ψ2.This includes the solution which is exponential inflation for ψo ＞ψ＞ψf , and then develops smoothly towards the radiation-like evolution for ψ＜ψf. The spectral indices of the scalar density ns and the gravitational wave fluctuations ng are computed. The value of ns lies well inside the limits set by the cosmic background explorer satellite.
Non-uniform Braneworld Stars: an Exact Solution
Ovalle, J
2008-01-01
The first exact interior solution to Einstein's field equations for a static and non-uniform braneworld star with local and non-local bulk terms is presented. It is shown that the bulk Weyl scalar ${\\cal U}(r)$ is always negative inside the stellar distribution, in consequence it reduces both the effective density and the effective pressure. It is found that the anisotropy generated by bulk gravity effect has an acceptable physical behaviour inside the distribution. Using a Reissner-N\\"{o}rdstrom-like exterior solution, the effects of bulk gravity on pressure and density are found through matching conditions.
Critical exact solutions for self-gravitating Dirac fields
Cianci, Roberto; Vignolo, Stefano
2016-01-01
We consider the Einstein-Dirac field equations describing a self-gravitating massive neutrino, looking for axially-symmetric exact solutions; in the search of general solutions, we find some that are specific and which have critical features, such as the fact that the space-time curvature turns out to be flat and the spinor field gives rise to a vanishing bi-linear scalar $\\overline{\\psi}\\psi=0$ with non-vanishing bi-linear pseudo-scalar $i\\overline{\\psi}\\gamma^5\\psi\
Institute of Scientific and Technical Information of China (English)
无
1998-01-01
The homogeneous balance method is extended to seek for B(a)cklund transformation, Lax pairs, non-local symmetries of variable coefficient KdV equation (VCKd VE).Then based on the B(a)cklund transformation and general solutions of a fourth-order nonlinear ordinary differential equation, five kinds of exact solutions of VCKdVE are derived. The soliton-like solution also belongs to these solutions.
F-expansion method and new exact solutions of the Schrödinger-KdV equation.
Filiz, Ali; Ekici, Mehmet; Sonmezoglu, Abdullah
2014-01-01
F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the Schrödinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number of Jacobi-elliptic function solutions are obtained including the Weierstrass-elliptic function solutions. When the modulus m of Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively. The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations in mathematical physics.
Conformal invariance and new exact solutions of the elastostatics equations
Chirkunov, Yu. A.
2017-03-01
We fulfilled a group foliation of the system of n-dimensional (n ≥ 2) Lame equations of the classical static theory of elasticity with respect to the infinite subgroup contained in normal subgroup of main group of this system. It permitted us to move from the Lame equations to the equivalent unification of two first-order systems: automorphic and resolving. We obtained a general solution of the automorphic system. This solution is an n-dimensional analogue of the Kolosov-Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. It turned out that in the two-dimensional and three-dimensional cases, which have a physical meaning, this system is conformally invariant, while the Lame equations admit only a group of similarities of the Euclidean space. This is a big success, since in the method of group foliation, resolving equations usually inherit Lie symmetries subgroup of the full symmetry group that was not used for the foliation. In the three-dimensional case for the solutions of the resolving system, we found the general form of the transformations similar to the Kelvin transformation. These transformations are the consequence of the conformal invariance of the resolving system. In the three-dimensional case with a help of the complex dependent and independent variables, the resolving system is written as a simple complex system. This allowed us to find non-trivial exact solutions of the Lame equations, which direct for the Lame equations practically impossible to obtain. For this complex system, all the essentially distinct invariant solutions of the maximal rank we have found in explicit form, or we reduced the finding of those solutions to the solving of the classical one-dimensional equations of the mathematical physics: the heat equation, the telegraph equation, the Tricomi equation, the generalized Darboux equation, and other equations. For the resolving system, we obtained double wave of a
Some exact BPS solutions for exotic vortices and monopoles
Ramadhan, Handhika S
2015-01-01
We present several analytical solutions of BPS vortices and monopoles in the generalized Abelian Maxwell-Higgs and Yang-Mills-Higgs theories, respectively. These models have recently been extensively studied and several exact solutions have already been obtained in~\\cite{Casana:2014qfa, Casana:2013lna}. In each theory, the dynamics is controlled by the additional two positive scalar-field-dependent functions, $f(|\\phi|)$ and $w(|\\phi|)$. For the case of vortices, we work in the ordinary symmetry-breaking Higgs potential, while for the case of monopoles we have the ordinary condition of the Prasad-Sommerfield limit. Our results generalize that of exact solutions found previously. We also present solutions for BPS vortices with higher winding number. These solutions suffer from the condition that $w(|\\phi|)$ has negative value at some finite range of $r$, but we argue that since it satisfies the weaker positive-value conditions then the corresponding energy density is still positive-definite and, thus, they are...
Generalized Nonlinear Proca Equation and its Free-Particle Solutions
Nobre, F D
2016-01-01
We introduce a non-linear extension of Proca's field theory for massive vector (spin $1$) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter $q$ (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit $q \\rightarrow 1$. We derive the nonlinear Proca equation from a Lagrangian that, besides the usual vectorial field $\\Psi^{\\mu}(\\vec{x},t)$, involves an additional field $\\Phi^{\\mu}(\\vec{x},t)$. We obtain exact time dependent soliton-like solutions for these fields having the...
Institute of Scientific and Technical Information of China (English)
丁皓江; 徐荣桥; 国凤林
1999-01-01
Based on three-dimensional elastic equations for piezoelectric materials, the state equations for piezoelectric circular plate under axisymmetric deformation are derived. Applying Hankel transform to them and letting the free boundary terms resulting from Hankel transform be zero, a set of ordinary differential equations with constant coefficients and associated boundary conditions are obtained. Furthermore, two exact solutions corresponding to generalized rigid slipping and generalized elastic simple support are deduced. Then, the governing equations obtained reduce to equations for axisymmetric problem of transversely isotropic circular plate. Under the two types of boundary conditions of elastic simple support and rigid slipping, exact solutions are derived. Finally, numerical results are presented and applicability of the classical plate theory is discussed.
Nonlinear Schrödinger equation from generalized exact uncertainty principle
Rudnicki, Łukasz
2016-09-01
Inspired by the generalized uncertainty principle, which adds gravitational effects to the standard description of quantum uncertainty, we extend the exact uncertainty principle approach by Hall and Reginatto (2002 J. Phys. A: Math. Gen. 35 3289), and obtain a (quasi)nonlinear Schrödinger equation. This quantum evolution equation of unusual form, enjoys several desired properties like separation of non-interacting subsystems or plane-wave solutions for free particles. Starting with the harmonic oscillator example, we show that every solution of this equation respects the gravitationally induced minimal position uncertainty proportional to the Planck length. Quite surprisingly, our result successfully merges the core of classical physics with non-relativistic quantum mechanics in its extremal form. We predict that the commonly accepted phenomenon, namely a modification of a free-particle dispersion relation due to quantum gravity might not occur in reality.
Travelling wave solutions for ( + 1)-dimensional nonlinear evolution equations
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2010-10-01
In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of ( + 1)-dimensional nonlinear evolution equations. Four models, the ( + 1)-dimensional generalized Boussinesq equation, ( + 1)-dimensional sine-cosine-Gordon equation, ( + 1)-double sinh-Gordon equation and ( + 1)-sinh-cosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling wave solutions are derived.
Institute of Scientific and Technical Information of China (English)
Chandaneswar Midya
2012-01-01
An analytical study of the distribution of a reactant solute undergoing a first-order chemical reaction in the boundary layer flow of an electrically conducting incompressible Buid over a linearly shrinking surface is presented. The Row is permeated by an externally applied magnetic Geld normal to the plane of the flow. The equations governing the Row and concentration Reid are reduced into a set of nonlinear ordinary differential equations using similarity variables. Closed form exact solutions of the reduced concentration equation are obtained for both prescribed power-law surface concentration (PSC) and power-law wall mass flux (PMF) as boundary conditions. The study reveals that the concentration over a shrinking sheet is signiRcantly different from that of a stretching surface. It s found that te solute boundary layer thickness is enhanced with the increasing values of the Schmidt number and the power-law index parameter, but decreases with enhanced vaJues of magnetic and reaction rate parameters for the PSC case. For the PMF case, the solute boundary layer thickness decreases with the increase of the Schmidt number, magnetic and reaction rate parameter for power-law index parameter n = 0. Negative solute boundary layer thickness is observed for the PMF case when n = 1 and 2, and these facts may not be realized in real-world applications.%An analytical study of the distribution of a reactant solute undergoing a first-order chemical reaction in the boundary layer flow of an electrically conducting incompressible fluid over a linearly shrinking surface is presented.The flow is permeated by an externally applied magnetic field normal to the plane of the flow.The equations governing the flow and concentration field are reduced into a set of nonlinear ordinary differential equations using similarity variables.Closed form exact solutions of the reduced concentration equation are obtained for both prescribed power-law surface concentration (PSC) and power-law wall
Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.
Islam, S M Rayhanul; Khan, Kamruzzaman; Akbar, M Ali
2015-01-01
In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.
Exact solutions of three dimensional black holes: Einstein gravity vs F(R) gravity
Hendi, S H; Saffari, R
2014-01-01
In this paper, we consider Einstein gravity in the presence of a class of nonlinear electrodynamics, called power Maxwell invariant (PMI). We take into account $(2+1)$-dimensional spacetime in Einstein-PMI gravity and obtain its black hole solutions. Then, we regard pure $F(R)$ gravity as well as $F(R)$-conformally invariant Maxwell theory to obtain exact solutions of the field equations with black hole interpretation. Finally, we investigate the conserved and thermodynamic quantities and discuss about the first law of thermodynamics for the mentioned gravitational models.
Directory of Open Access Journals (Sweden)
Zayed EL Sayed M.E.
2016-01-01
Full Text Available In this article, we apply the generalized Kudryashov method for finding exact solutions of three nonlinear partial differential equations (PDEs, namely: the Biswas-Milovic equation with dual-power law nonlinearity; the Zakharov--Kuznetsov equation (ZK(m,n,k; and the K(m,n equation with the generalized evolution term. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, and hyperbolic function solutions. Physical explanations for certain solutions of the three nonlinear PDEs are obtained.
Energy Technology Data Exchange (ETDEWEB)
Chen, Yong [Ningbo Univ., Ningbo (China). Department of Mathematics; Shanghai Jiao-Tong Univ., Shangai (China). Department of Physics; Chinese Academy of sciences, Beijing (China). Key Laboratory of Mathematics Mechanization
2005-03-01
A general method to uniformly construct exact solutions in terms of special function of nonlinear partial differential equations is presented by means of a more general ansatz and symbolic computation. Making use of the general method, we can successfully obtain the solutions found by the method proposed by Fan (J. Phys. A., 36 (2003) 7009) and find other new and more general solutions, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solution, soliton solutions, soliton-like solutions and Jacobi, Weierstrass doubly periodic wave solutions. A general variable-coefficient two-dimensional KdV equation is chosen to illustrate the method. As a result, some new exact soliton-like solutions are obtained. planets. The numerical results are given in tables. The results are discussed in the conclusion.
Exact Spherical Wave Solutions to Maxwell's Equations with Applications
Silvestri, Guy G.
Electromagnetic radiation from bounded sources represent an important class of physical problems that can be solved for exactly. However, available texts on this subject almost always resort to approximate solution techniques that not only obscure the essential features of the problem but also restrict application to limited ranges of observation. This dissertation presents exact solutions for this important class of problems and demonstrates how these solutions can be applied to situations of genuine physical interest, in particular, the design of device structures with prespecified emission characteristics. The strategy employed is to solve Maxwell's equations in the spherical coordinate system. In this system, fundamental parameters such as electric and magnetic multipole moments fall out quite naturally. Expressions for radiated power, force, and torque assume especially illuminating and simple forms when expressed in terms of these multipole moments. All solutions are derived ab initio using first-principles arguments exclusively. Two operator-equations that receive particularly detailed treatment are the vector Helmholtz equation for the time-independent potential vec a and the "covariant divergence" equation for the energy-momentum-stress tensor T^{mu nu}. An application of classical formulas, as modified by the requirements of statistical mechanics, to the case of heated blackbodies leads to inquiries into the foundations of quantum mechanics and their relation to classical field theory. An application of formulas to various emission structures (spherically-shaped antennas, surface diffraction gratings, collimated beams) provides a basis upon which to characterize these structures in an exact sense, and, ultimately, to elicit clues as to their optimum design.
Optimum intermediate fibers for reducing interconnection loss: exact solution.
Yablon, Andrew D; Sumetsky, M
2007-03-15
We derive an exact analytical solution for a transmission line of N single-mode intermediate optical fibers that minimize the interconnection loss between any two dissimilar fiber modes that are well described by that paraxial scalar wave equation. Our solution shows that N optimum intermediate fibers reduce the original interconnection loss by a factor of least 1/(N+1) and that the total interconnection loss is only a function of N and the original direct interconnection loss. Our solution is not restricted to axisymmetric fibers or mode fields and therefore could be useful for reducing the interconnection loss between microstructured optical fibers, between certain slab waveguides, or between fibers and optical sources or detectors.
Saha Ray, S.; Sahoo, S.
2017-01-01
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely time fractional modified Kawahara equations by using the ( G^'/G)-expansion method via fractional complex transform. As a result, new types of exact analytical solutions are obtained.
Ferrofluid patterns in Hele-Shaw cells: Exact, stable, stationary shape solutions
Lira, Sergio; Miranda, Jose
2016-11-01
We investigate a quasi-two-dimensional system composed by an initially circular ferrofluid droplet surrounded by a nonmagnetic fluid of higher density. These immiscible fluids flow in a rotating Hele-Shaw cell, under the influence of an in-plane radial magnetic field. We focus on the situation in which destabilizing bulk magnetic field effects are balanced by stabilizing centrifugal forces. In this framing, we consider the interplay of capillary and magnetic normal traction effects in determining the fluid-fluid interface morphology. By employing a vortex-sheet formalism we have been able to find a family of exact stationary N-fold polygonal shape solutions for the interface. A weakly nonlinear theory is then used to verify that such exact interfacial solutions are in fact stable. We thank CNPq (Brazilian Research Council) for financial support.
Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities
Pınar, Zehra; Deliktaş, Ekin
2017-02-01
The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time- space modulated nonlinearities of coupled Schrödinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.
Exact Solution in Chaotic Inflation Model with Potential Minima
Institute of Scientific and Technical Information of China (English)
WANG Wen-Fu
2001-01-01
Taking the cosmological expansion rate directly as a function of field , H = H( ), we present a new exact solution to Einstein's equations that describe the evolution of cosmological chaotic inflation model. The inflation is driven by the evolution of scalar field with inflation potential V( ) = λ 2 v2)2.8 ( 2- 2)2.The spectral indices of the scalar density ns and gravitational wave fluctuations ng are computed. The value of ns lies well inside the limits set by the cosmic background explorer satellite.
A class of exact classical solutions to string theory.
Coley, A A
2002-12-31
We show that the recently obtained class of spacetimes for which all of the scalar curvature invariants vanish (which can be regarded as generalizations of pp-wave spacetimes) are exact solutions in string theory to all perturbative orders in the string tension scale. As a result the spectrum of the theory can be explicitly obtained, and these spacetimes are expected to provide some hints for the study of superstrings on more general backgrounds. Since these Lorentzian spacetimes suffer no quantum corrections to all loop orders they may also offer insights into quantum gravity.
Exact Inflationary Solution in String-Motivated Scalar Field Cosmology
Institute of Scientific and Technical Information of China (English)
王文福
2001-01-01
We present a new exact solution to Einstein's equations that describes the evolution of inflationary universe models. The inflation is driven by the evolution of a scalar field with an approximate two-loop four-dimensional string potential. In this scenario, the inflation began immediately after the epoch governed by quantum gravity and therefore there is no initial singularity. The successful inflation scenario is expected to appear only at two loop order. For a1/｜a2｜ ≥ 90, the spectral index ns of the scalar density fluctuations lies well inside the limits set by the cosmic background explorer satellite and the gravitational wave spectral index is ng≈1.
Exact Solution of a Drop-Push Model for Percolation
Majumdar, Satya N.; Dean, David S.
2002-08-01
Motivated by a computer science algorithm known as ``linear probing with hashing,'' we study a new type of percolation model whose basic features include a sequential ``dropping'' of particles on a substrate followed by their transport via a ``pushing'' mechanism. Our exact solution in one dimension shows that, unlike the ordinary random percolation model, the drop-push model has nontrivial spatial correlations generated by the dynamics itself. The critical exponents in the drop-push model are also different from those of the ordinary percolation. The relevance of our results to computer science is pointed out.
Exact solutions and ladder operators for a new anharmonic oscillator
Energy Technology Data Exchange (ETDEWEB)
Dong Shihai [Programa de Ingenieria Molecular, Instituto Mexicano del Petroleo, Lazaro Cardenas 152, 07730 Mexico DF (Mexico)]. E-mail: dongsh2@yahoo.com; Sun Guohua [Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, UNAM, A.P. 20-726, Del. Alvaro Obregon, 01000 Mexico DF (Mexico); Lozada-Cassou, M. [Programa de Ingenieria Molecular, Instituto Mexicano del Petroleo, Lazaro Cardenas 152, 07730 Mexico DF (Mexico)
2005-06-06
In this Letter, we propose a new anharmonic oscillator and present the exact solutions of the Schrodinger equation with this oscillator. The ladder operators are established directly from the normalized radial wave functions and used to evaluate the closed expressions of matrix elements for some related functions. Some comments are made on the general calculation formula and recurrence relation for off-diagonal matrix elements. Finally, we show that this anharmonic oscillator possesses a hidden symmetry between E(r) and E(ir) by substituting r->ir.
Exact solutions and conserved quantities in f( R, T) Gravity
Sharif, M.; Nawazish, Iqra
2017-06-01
This paper explores Noether and Noether gauge symmetries of anisotropic universe model in f( R, T) gravity. We consider two particular models of this gravity and evaluate their symmetry generators as well as associated conserved quantities. We also find exact solution by using cyclic variable and investigate its behavior via cosmological parameters. The behavior of cosmological parameters turns out to be consistent with recent observations which indicates accelerated expansion of the universe. Next we study Noether gauge symmetry and corresponding conserved quantities for both isotropic and anisotropic universe models. We conclude that symmetry generators and the associated conserved quantities appear in all cases.
Exact solutions for the spin tune for model storage rings
Mane, S R
2002-01-01
We present exact analytical expressions for the spin tune for arbitrary values of the orbital action for several storage ring models. The models we treat contain Siberian Snakes, the use of which is essential to preserve the polarization of beams in high-energy proton storage rings. Our solutions contain some novel features. We also prove a previously conjectured claim about the behavior of spin tuneshifts in rings with multiple Snakes. The conjecture is based on numerical simulations, but our proof is analytical, and also nonperturbative.
Piezoelectric vibration damping using resonant shunt circuits: an exact solution
Soltani, P.; Kerschen, G.; Tondreau, G.; Deraemaeker, A.
2014-12-01
The objective of this paper is to propose an exact closed-form solution to the {{H}∞ } optimization of piezoelectric materials shunted with inductive-resistive passive electrical circuits. Realizing that Den Hartog's method which imposes fixed points of equal height in the receptance transfer function is approximate, the parameters of the piezoelectric tuned vibration absorber are calculated through the direct minimization of the maxima of the receptance. The method is applied to a one-degree-of-freedom primary oscillator considering various values of the electromechanical coupling coefficients.
Scattering of two photons from two distant qubits: exact solution
Energy Technology Data Exchange (ETDEWEB)
Laakso, Matti; Pletyukhov, Mikhail [Institute for Theory of Statistical Physics, RWTH Aachen, 52056 Aachen (Germany)
2015-07-01
We consider the inelastic scattering of two photons from two qubits separated by an arbitrary distance and coupled to a one-dimensional transmission line. We present an exact, analytical solution to the problem, and use it to explore a particular configuration of qubits which is transparent to single-photon scattering, thus highlighting non-Markovian effects of inelastic two-photon scattering: Strong two-photon interference and momentum dependent photon (anti)bunching. This latter effect can be seen as an inelastic generalization of the Hong-Ou-Mandel effect.
Exact solution of an evolutionary model without aging
Onody, Roberto N.; de Medeiros, Nazareno G. F.
1999-09-01
We introduce an age-structured asexual population model containing all the relevant features of evolutionary aging theories. Beneficial as well as deleterious mutations, heredity, and arbitrary fecundity are present and managed by natural selection. An exact solution without aging is found. We show that fertility is associated with generalized forms of the Fibonacci sequence, while mutations and natural selection are merged into an integral equation which is solved by Fourier series. Average survival probabilities and Malthusian growth exponents are calculated and indicate that the system may exhibit mutational meltdown. The relevance of the model in the context of fissile reproduction groups like many protozoa and coelenterates is discussed.
Institute of Scientific and Technical Information of China (English)
Zha Qi-Lao; Sirendaoreji
2006-01-01
Some new exact solitary wave solutions of the Hybrid lattice and discrete mKdV lattice are obtained by using a hyperbolic function approach.This approach can also be applied to other nonlinear differential-difference equations.
Exact solutions to model surface and volume charge distributions
Mukhopadhyay, S.; Majumdar, N.; Bhattacharya, P.; Jash, A.; Bhattacharya, D. S.
2016-10-01
Many important problems in several branches of science and technology deal with charges distributed along a line, over a surface and within a volume. Recently, we have made use of new exact analytic solutions of surface charge distributions to develop the nearly exact Boundary Element Method (neBEM) toolkit. This 3D solver has been successful in removing some of the major drawbacks of the otherwise elegant Green's function approach and has been found to be very accurate throughout the computational domain, including near- and far-field regions. Use of truly distributed singularities (in contrast to nodally concentrated ones) on rectangular and right-triangular elements used for discretizing any three-dimensional geometry has essentially removed many of the numerical and physical singularities associated with the conventional BEM. In this work, we will present this toolkit and the development of several numerical models of space charge based on exact closed-form expressions. In one of the models, Particles on Surface (ParSur), the space charge inside a small elemental volume of any arbitrary shape is represented as being smeared on several surfaces representing the volume. From the studies, it can be concluded that the ParSur model is successful in getting the estimates close to those obtained using the first-principles, especially close to and within the cell. In the paper, we will show initial applications of ParSur and other models in problems related to high energy physics.
Directory of Open Access Journals (Sweden)
M. Fazli Aghdaei
2011-09-01
Full Text Available In this work, the generalized ( G0 G –expansion method is applied to solve the coupled Boiti-Leon-Pempinelli system (BLP system. Generalized ( G0 G -expansion method was used to construct solitary wave solutions of the nonlinear evolution equations. This method is developed for searching the exact travelling wave solutions of nonlinear partial differential equations. It is shown that the ( G0 G -expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear partial differential equations
Critical exact solutions for self-gravitating Dirac fields
Energy Technology Data Exchange (ETDEWEB)
Cianci, Roberto; Fabbri, Luca; Vignolo, Stefano [Universita di Genova, DIME Sez. Metodi e Modelli Matematici, Genova (Italy)
2016-11-15
We consider the Einstein-Dirac field equations describing a self-gravitating massive neutrino, looking for axially symmetric exact solutions; in the search of general solutions, we find some that are specific and which have critical features, such as the fact that the space-time curvature turns out to be flat and the spinor field gives rise to a vanishing bi-linear scalar ψψ =0 with non-vanishing bi-linear pseudo-scalar iψγ{sup 5}ψ ≠ 0: because in quantum-field theory general computational methods are built on plane-wave solutions, for which the bi-linear pseudo-scalar vanishes while the bi-linear scalar does not vanish, then the solutions we found cannot be treated with the usual machinery of quantum-field theory. This means that for the Einstein-Dirac system there exist admissible solutions which nevertheless cannot be quantized with the common prescriptions; we regard this situation as yet another issue of tension between Einstein gravity and quantum principles. Possible ways to quench this tension can be seen either in enlarging the validity of quantum-field theory or by restricting the space of the solutions of the Einstein-Dirac system of field equations. (orig.)
Exact Cosmological Solutions in Modified Brans--Dicke Theory
Rasouli, S M M
2016-01-01
In this paper, we obtain exact cosmological vacuum solutions for an extended FLRW homogenous and isotropic Brans-Dicke (BD) universe in five dimensions for all values of the curvature index. Then, by employing the equations associated to a modified Brans-Dicke theory (MBDT) [1], we construct the physics on a four-dimensional hypersurface. We show that the induced matter obeys the equation of state of a fluid of a barotropic type. We discuss the properties of such an induced matter for some values of the equation of state parameter and analyze in detail their corresponding solutions. To illustrate the cosmological behaviors of the solutions, we contrast our solutions with those present the standard Brans-Dicke theory. We retrieve that, in MBDT scenario, it is impossible to find a physically acceptable solution associated to the negative curvature for both the dust-dominated and radiation-dominated universes. However, for a spatially flat and closed universes, we argue that our obtained solutions are more gener...
A New Rational Algebraic Approach to Find Exact Analytical Solutions to a (2+1)-Dimensional System
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions.The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions,and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on (2+1)-dimensional dispersive long-wave equation.
Directory of Open Access Journals (Sweden)
Juan Carlos Ceballos V.
2005-10-01
Full Text Available The exact boundary controllability of the higher order nonlinear Schrodinger equation with constant coefficients on a bounded domain with various boundary conditions is studied. We derive the exact boundary controllability for this equation for sufficiently small initial and final states.
Yang, Xiao-Feng; Deng, Zi-Chen; Li, Qing-Jun; Wei, Yi
2016-07-01
The homogeneous balance of undetermined coefficients method (HBUCM) is firstly proposed to construct not only the exact traveling wave solutions, three-wave solutions, homoclinic solutions, N-soliton solutions, but also multi-symplectic structures of some nonlinear partial differential equations (NLPDEs). By applying the proposed method to the variant Boussinesq equations (VBEs), the exact combined traveling wave solutions and a multi-symplectic structure of the VBEs are obtained directly. Then, the definition and a multi-symplectic structure of the variant Boussinesq-Whitham-Broer-Kaup type equations (VBWBKTEs) which can degenerate to the VBEs, the Whitham-Broer-Kaup equations (WBKEs) and the Broer-Kaup equations (BKEs) are given in the multi-symplectic sense. The HBUCM is also a standard and computable method, which can be generalized to obtain the exact solutions and multi-symplectic structures for some types of NLPDEs.
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Ahmad Neirameh
2016-01-01
Full Text Available This paper obtains solutions as well as other solutions to the 3D- Gross–Pitaevskii equation, which is called the non-linear Schrodinger equation under the conditions of Kudryashov method that appear in various areas of mathematical physics. This equation describes Bose–Einstein condensates in the low temperature regime. These new exact solutions will complement previous results and help further to understand the physical structures.
Auxiliary Equation Method and New Exact Solutions of BKP Equation%任意方程法及BKP方程新的精确解
Institute of Scientific and Technical Information of China (English)
马红彩; 张娅莉; 邓爱平
2008-01-01
In this paper,auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation.We study the(2+1)-dimensional BKP equation and get a series of new types of traveling wave solutions.The method used here can be also extended to other nonlinear partial differential equations.
Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
Cordova, Clay; Popolitov, Alexandr; Shakirov, Shamil
2016-01-01
We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model. We extend our formulas to include both logarthmic and finite-difference deformations, thereby generalizing the celebrated Selberg and Kadell integrals. We conjecture a formula for correlators of two Schur functions in these models, and explain how our results follow from a general orbifold-like procedure that can be applied to any one-matrix model with a single-trace potential.
Exact solutions and physical analogies for unidirectional flows
Bazant, Martin Z
2016-01-01
Unidirectional flow, exemplified by Hagen-Poiseuille flow in a circular pipe, is the simplest phenomenon of fluid mechanics. Its mathematical description, the Dirichlet problem for Poisson's equation in two dimensions with constant forcing, arises in many physical contexts, such as the torsion of elastic beams, first solved by de Saint-Venant and extended to arbitrary domains by Muskhelishvilli. In this article, we unify and extend the literature by identifying fifteen physical analogies for unidirectional flow and describing their common mathematical structure. We show how to construct approximate geometries that admit exact solutions, by adding harmonic functions to quadratic forms or truncating eigenfunction expansions, and we discuss the difficulty of the evaluating the general contour-integral solution for a given geometry. We prove that the remarkable geometrical interpretation of Poiseuille flow in an equilateral pipe, as the product of the distances from an interior point to the sides, is only shared ...
Scalar triplet on a domain wall: an exact solution
Gani, Vakhid A.; Lizunova, Mariya A.; Radomskiy, Roman V.
2016-04-01
We study a model with a real scalar Higgs field and a scalar triplet field that allows existence of a topological defect — a domain wall. The wall breaks the global O(3) symmetry of the model, which gives rise to non-Abelian orientational degrees of freedom. We found an exact analytic solution that describes a domain wall with a localized configuration of the triplet field on it. This solution enables one to calculate contributions to the action from the orientational and translational degrees of freedom of the triplet field. We also study the linear stability of the domain wall with the triplet field switched off. We obtain that degrees of freedom localized on the wall can appear or do not appear depending on the parameters of the model.
Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Exact solutions and physical analogies for unidirectional flows
Bazant, Martin Z.
2016-06-01
Unidirectional flow is the simplest phenomenon of fluid mechanics. Its mathematical description, the Dirichlet problem for Poisson's equation in two dimensions with constant forcing, arises in many physical contexts, such as the torsion of elastic beams, first solved by de Saint-Venant for complex shapes. Here the literature is unified and extended by identifying 17 physical analogies for unidirectional flow and describing their common mathematical structure. Besides classical analogies in fluid and solid mechanics, applications are discussed in stochastic processes (first passage in two dimensions), pattern formation (river growth by erosion), and electrokinetics (ion transport in nanochannels), which also involve Poisson's equation with nonconstant forcing. Methods are given to construct approximate geometries that admit exact solutions, by adding harmonic functions to quadratic forms or by truncating eigenfunction expansions. Exact solutions for given geometries are also derived by conformal mapping. We prove that the remarkable geometrical interpretation of Poiseuille flow in an equilateral triangular pipe (the product of the distances from an interior point to the sides) is only shared by parallel plates and unbounded equilateral wedges (with the third side hidden behind the apex). We also prove Onsager reciprocity for linear electrokinetic phenomena in straight pores of arbitrary shape and surface charge, based on the mathematics of unidirectional flow.
Generalized Jacobi Elliptic Function Solution to a Class of Nonlinear Schrödinger-Type Equations
Directory of Open Access Journals (Sweden)
Zeid I. A. Al-Muhiameed
2011-01-01
Full Text Available With the help of the generalized Jacobi elliptic function, an improved Jacobi elliptic function method is used to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation are investigated, and the exact solutions are derived with the aid of the homogenous balance principle.
Lie group classification and exact solutions of the generalized Kompaneets equations
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Oleksii Patsiuk
2015-04-01
Full Text Available We study generalized Kompaneets equations (GKEs with one functional parameter, and using the Lie-Ovsiannikov algorithm, we carried out the group classification. It is shown that the kernel algebra of the full groups of the GKEs is the one-dimensional Lie algebra. Using the direct method, we find the equivalence group. We obtain six non-equivalent (up to transformations from the equivalence group GKEs that allow wider invariance algebras than the kernel one. We find a number of exact solutions of the non-linear GKE which has the maximal symmetry properties.
Exact solution of the pairing problem for spherical and deformed systems
Qi, Chong
2016-01-01
There has been increasing interest in studying the Richardson model from which one can derive the exact solution for certain pairing Hamiltonians. However, it is still a numerical challenge to solve the nonlinear equations involved. In this paper we tackle this problem by employing a simple hybrid polynomial approach. The method is found to be robust and is valid for both deformed and nearly spherical nuclei. It also provides important and convenient initial guesses for spherical systems with large degeneracy. As an example, we apply the method to study the shape coexistence in neutron-rich Ni isotopes.
Comparison of Exact Solutions for Heat Transfer in Extended Surfaces of Different Geometries
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K. J. Moleofane
2014-01-01
nonlinear steady state problem is linearizable provided that the thermal conductivity is the differential consequence of the term involving the heat transfer coefficient. As such, one is able to construct exact solutions. On the other hand, we employ the Lie point symmetry methods when the problem is not linearizable. Some interesting results are obtained and analyzed. The effects of the parameters such as thermogeometric fin parameter and the exponent on temperature are studied. Furthermore, fin efficiency and heat flux along the fin length of a spherical geometry are also studied.
Institute of Scientific and Technical Information of China (English)
WANG Qi; CHEN Yong; LI Biao; ZHANG Hong-Qing
2004-01-01
Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e.,the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.
Localized light waves: Paraxial and exact solutions of the wave equation (a review)
Kiselev, A. P.
2007-04-01
Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “ X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.
Solution and Positive Solution to Nonlinear Cantilever Beam Equations
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using the decomposition technique of equation and the fixed point theorem, the existence of solution and positive solution is studied for a nonlinear cantilever beam equation. The equation describes the deformation of the elastic beam with a fixed end and a free end. The main results show that the equation has at least one solution or positive solution, provided that the "height" of nonlinear term is appropriate on a bounded set.
Properties of exact solution of second-order differential equation with pantograph delay
Institute of Scientific and Technical Information of China (English)
李冬松; 白红; 刘明珠
2002-01-01
This paper is concerned with properties of exact solution of pantograph delay equation y′′(t)＝ay′(t)+by(t)+cy(qt), 0＜q＜1. Firstly, the existence and uniqueness of the exact solution of equations are proved, and then the condition is investigated which guarantee the exact solution is asymptotic stable.
Some Exact Solutions of Magnetized viscous model in String Cosmology
Singh, C P
2012-01-01
In this paper we study anisotropic Bianchi-V universe with magnetic field and bulk viscous fluid in string cosmology. Exact solutions of the field equations are obtained by using the equation of state for a cloud of strings and a relationship between bulk viscous coefficient and expansion scalar. The bulk viscous coefficient is assumed to be inversely proportional to the expansion scalar. It is interesting to examine the effects of magnetized bulk viscous string model in early and late stages of the evolution of the universe. This paper investigates the different string models like geometrical(Nambu string), Takabayashi (p-string) and Reddy string models by taking certain physical conditions. The introduction of magnetic field or bulk viscosity or both results in rapid change in scale factors as well as in the classical potential. The presence of viscosity prevents the universe to be empty in its future evolution. The physical and geometrical aspects of each string model are discussed in detail.
Exact solutions for hydrodynamic interactions of two squirming spheres
Papavassiliou, Dario
2016-01-01
We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest...
Kink wave determined by parabola solution of a nonlinear ordinary differential equation
Institute of Scientific and Technical Information of China (English)
LI Ji-bin; LI Ming; NA Jing
2007-01-01
By finding a parabola solution connecting two equilibrium points of a planar dynamical system, the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown. Some exact explicit parametric representations of kink wave solutions are given. Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.
Energy Technology Data Exchange (ETDEWEB)
Liu Chunping
2003-06-02
Using a direct algebraic method, more new exact solutions of the Kolmogorov-Petrovskii-Piskunov equation are presented by formula form. Then a theorem concerning the relation between the kink-type solution and the kink-bell-type solution of nonlinear evolution equations is given. Finally, the applications of the theorem to several well-known equations in physics are also discussed.
Applications of algebraic method to exactly solve some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)]. E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)]. E-mail: aramady@yahoo.com
2007-08-15
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear evolution equations is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDE's) are obtained. Graphs of the solutions are displayed.
Institute of Scientific and Technical Information of China (English)
许涛
2004-01-01
M r.Sm ith liked to be exact. O ne day when he was w alking in thestreet a m an cam e over and asked him E xcuse m e but w here's the , : “ ,nearest bookshop ?” The nearest bookshop Y ou have to cross a bridge and then turn “ ?to the right. ” A nd is the bridge long “ ?” Thirty m eters. “ ” The m an thanked him and went towards the bridge. Suddenly heheard som eone running after him . Stop M r.Sm ith w as shouting. I'm sorry. I just rem em bered ...
Hayek, Mohamed
2014-09-01
The paper presents certain exact solutions describing the vertical movement of a water pulse through a semi-infinite unsaturated porous column. The saturation-based form of the Richards' equation is used with special power law relative-permeability functions. Both capillary and gravity effects are taken into account. Three exact solutions are derived corresponding to three relative-permeability functions, linear, quadratic and cubic. The Richards' equation is nonlinear for the three cases. The solutions are obtained by applying a general similarity transformation. They are explicit in space and time variables and do not contain any approximation. They describe the evolution of the water saturation in the vertical column and they can be used to predict the post-infiltration movement of a finite quantity of water. Exact expressions of the masses of water leaving a given depth are also derived for the three cases. We analyze the effect of relative-permeability and capillary pressure. The proposed solutions are also useful for checking numerical schemes. One of the exact solutions is used to validate numerical solution obtained from an arbitrary initial condition. Results show that the numerical solution converges to the exact solution for large times.
Exact Solutions for Stokes' Flow of a Non-Newtonian Nanofluid Model: A Lie Similarity Approach
Aziz, Taha; Aziz, A.; Khalique, C. M.
2016-07-01
The fully developed time-dependent flow of an incompressible, thermodynamically compatible non-Newtonian third-grade nanofluid is investigated. The classical Stokes model is considered in which the flow is generated due to the motion of the plate in its own plane with an impulsive velocity. The Lie symmetry approach is utilised to convert the governing nonlinear partial differential equation into different linear and nonlinear ordinary differential equations. The reduced ordinary differential equations are then solved by using the compatibility and generalised group method. Exact solutions for the model equation are deduced in the form of closed-form exponential functions which are not available in the literature before. In addition, we also derived the conservation laws associated with the governing model. Finally, the physical features of the pertinent parameters are discussed in detail through several graphs.
Institute of Scientific and Technical Information of China (English)
LIU Chun-Ping; LING Zhi
2005-01-01
By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.
Institute of Scientific and Technical Information of China (English)
G. Darmani; S. Setayeshi; H. Ramezanpour
2012-01-01
In this paper an efficient computational method based on extending the sensitivity approach （SA） is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations （DDEs）, the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach.
Multiple scales analysis and travelling wave solutions for KdV type nonlinear evolution equations
Ayhan, Burcu; Ozer, M. Naci; Bekir, Ahmet
2017-01-01
Nonlinear evolution equations are the mathematical models of problems that arise in many field of science. These equations has become an important field of study in applied mathematics in recent years. We apply exact solution methods and multiple scale method which is known as a perturbation method to nonlinear evolution equations. Using exact solution methods we get travelling wave solutions expressed by hyperbolic functions, trigonometric functions and rational functions. Also we derive Nonlinear Schrödinger (NLS) type equations from Korteweg-de Vries (KdV) type nonlinear evolution equations and we get approximate solutions for KdV type equations using multiple scale method. The proposed methods are direct and effective and can be used for many nonlinear evolution equations. It is shown that these methods provide a powerful mathematical tool to solve nonlinear evolution equations in mathematical physics.
Directory of Open Access Journals (Sweden)
Roman Cherniha
2016-06-01
Full Text Available The nonlinear mathematical model for solute and fluid transport induced by the osmotic pressure of glucose and albumin with the dependence of several parameters on the hydrostatic pressure is described. In particular, the fractional space available for macromolecules (albumin was used as a typical example and fractional fluid void volume were assumed to be different functions of hydrostatic pressure. In order to find non-uniform steady-state solutions analytically, some mathematical restrictions on the model parameters were applied. Exact formulae (involving hypergeometric functions for the density of fluid flux from blood to tissue and the fluid flux across tissues were constructed. In order to justify the applicability of the analytical results obtained, a wide range of numerical simulations were performed. It was found that the analytical formulae can describe with good approximation the fluid and solute transport (especially the rate of ultrafiltration for a wide range of values of the model parameters.
Institute of Scientific and Technical Information of China (English)
Bai Cheng-Lin; Zhang Xia; Zhang Li-Hua
2009-01-01
This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential-differenceequations (NLDDEs) and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the (2+l)-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential-difference equations.
Logical gaps in the approximate solutions of the social learning game and an exact solution.
Dai, Wenjie; Wang, Xin; Di, Zengru; Wu, Jinshan
2014-01-01
After the social learning models were proposed, finding solutions to the games becomes a well-defined mathematical question. However, almost all papers on the games and their applications are based on solutions built either upon an ad-hoc argument or a twisted Bayesian analysis of the games. Here, we present logical gaps in those solutions and offer an exact solution of our own. We also introduce a minor extension to the original game so that not only logical differences but also differences in action outcomes among those solutions become visible.
Logical gaps in the approximate solutions of the social learning game and an exact solution.
Directory of Open Access Journals (Sweden)
Wenjie Dai
Full Text Available After the social learning models were proposed, finding solutions to the games becomes a well-defined mathematical question. However, almost all papers on the games and their applications are based on solutions built either upon an ad-hoc argument or a twisted Bayesian analysis of the games. Here, we present logical gaps in those solutions and offer an exact solution of our own. We also introduce a minor extension to the original game so that not only logical differences but also differences in action outcomes among those solutions become visible.
Dual solution of Casson fluid over a porous medium: Exact solutions with extra boundary condition
Khan, Najeeb Alam; Khan, Sidra
2016-12-01
In this article we calculate the exact solution of the steady flow of non-Newtonian Casson fluid, over a stretching/shrinking sheet. The governing partial differential equations (PDEs) are transformed into ordinary differential equation (ODE) by using similarity transformation and then solved analytically by utilizing the exact solution. The closed form unique solution is obtained in the case of stretching sheet whereas for shrinking sheet unique and dual solutions are obtained. Influences of Casson fluid and suction/injection parameter on dimensionless velocity function are discussed and plotted graphically; also the effects of skin friction coefficient are presented in graphical form. Comparisons of current solutions with previous study are also made for the verification of the present study.
Nonlinear robust control of proton exchange membrane fuel cell by state feedback exact linearization
Energy Technology Data Exchange (ETDEWEB)
Li, Q.; Chen, W. [School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, Sichuan Province (China); Wang, Y.; Jia, J. [School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue 639798, Singapore (Singapore); Han, M. [School of Engineering, Temasek Polytechnic, Tampines 529757, Singapore (Singapore)
2009-10-20
By utilizing the state feedback exact linearization approach, a nonlinear robust control strategy is designed based on a multiple-input multiple-output (MIMO) dynamic nonlinear model of proton exchange membrane fuel cell (PEMFC). The state feedback exact linearization approach can achieve the global exact linearization via the nonlinear coordinate transformation and the dynamic extension algorithm such that H{sub {infinity}} robust control strategy can be directly utilized to guarantee the robustness of the system. The proposed dynamic nonlinear model is tested by comparing the simulation results with the experimental data in Fuel Cell Application Centre in Temasek Polytechnic. The comprehensive results of simulation manifest that the dynamic nonlinear model with nonlinear robust control law has better transient and robust stability when the vehicle running process is simulated. The proposed nonlinear robust controller will be very useful to protect the membrane damage by keeping the pressure deviations as small as possible during large disturbances and prolong the stack life of PEMFC. (author)
Exact solution of the classical mechanical quadratic Zeeman effect
Indian Academy of Sciences (India)
Sambhu N Datta; Anshu Pandey
2007-06-01
We address the curious problem of quadratic Zeeman effect at the classical mechanical level. The problem has been very well understood for decades, but an analytical solution of the equations of motion is still to be found. This state of affairs persists because the simultaneous presence of the Coulombic and quadratic terms lowers the dynamical symmetry. Energy and orbital angular momentum are still constants of motion. We find the exact solutions by introducing the concept of an image ellipse. The quadratic effect leads to a dilation of space–time, and a one-to-one correspondence is observed for pairs of physical quantities like energy and angular momentum, and the maximum and minimum distances from the Coulomb center for the Zeeman orbit and the corresponding pairs for the image ellipse. Thus, instead of finding additional conserved quantities, we find constants of motion for an additional dynamics, namely, the image problem. The trajectory is open, in agreement with Bertrand's theorem, but necessarily bound. A stable unbound trajectory does not exist for real values of energy and angular momentum. The radial distance, the angle covered in the plane of the orbit, and the time are uniquely determined by introducing further the concept of an image circle. While the radial distance is defined in a closed form as a transcendental function of the image-circular angle, the corresponding orbit angle and time variables are found in the form of two convergent series expansions. The latter two variables are especially contracted, thereby leading to a precession of the open cycles around the Coulomb center. It is expected that the space–time dilation effect observed here would somehow influence the solution of the quantum mechanical problem at the non-relativistic level.
Nonlinear inertial oscillations of a multilayer eddy: An analytical solution
Dotsenko, S. F.; Rubino, A.
2008-06-01
Nonlinear axisymmetric oscillations of a warm baroclinic eddy are considered within the framework of an reduced-gravity model of the dynamics of a multilayer ocean. A class of exact analytical solutions describing pure inertial oscillations of an eddy formation is found. The thicknesses of layers in the eddy vary according to a quadratic law, and the horizontal projections of the velocity in the layers depend linearly on the radial coordinate. Owing to a complicated structure of the eddy, weak limitations on the vertical distribution of density, and an explicit form of the solution, the latter can be treated as a generalization of the exact analytical solutions of this form that were previously obtained for homogeneous and baroclinic eddies in the ocean.
Symmetrized solutions for nonlinear stochastic differential equations
Directory of Open Access Journals (Sweden)
G. Adomian
1981-01-01
Full Text Available Solutions of nonlinear stochastic differential equations in series form can be put into convenient symmetrized forms which are easily calculable. This paper investigates such forms for polynomial nonlinearities, i.e., equations of the form Ly+ym=x where x is a stochastic process and L is a linear stochastic operator.
Raslan, K. R.; EL-Danaf, Talaat S.; Ali, Khalid K.
2017-07-01
In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann-Liouville derivative. We plot the exact solutions for these equations at different time levels.
Energy Technology Data Exchange (ETDEWEB)
Kong Dexing [Department of Mathematics, Zhejiang University, Hangzhou 310027 (China); Sun Qingyou, E-mail: qysun@cms.zju.edu.cn [Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027 (China)
2011-04-01
All articles must In this paper we introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution which implies the exact controllability of this kind of wave equation. Furthermore, based on this, we study the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability in these cases. Finally, as an application, we introduce the hyperbolic curvature flow and obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves.
ANALYTICAL SOLUTION OF NONLINEAR BAROTROPIC VORTICITY EQUATION
Institute of Scientific and Technical Information of China (English)
WANG Yue-peng; SHI Wei-hui
2008-01-01
The stability of nonlinear barotropic vorticity equation was proved. The necessary and sufficient conditions for the initial value problem to be well-posed were presented. Under the conditions of well-posedness, the corresponding analytical solution was also gained.
GLOBAL SOLUTIONS OF NONLINEAR SCHRODINGER EQUATIONS
Institute of Scientific and Technical Information of China (English)
Ye Yaojun
2005-01-01
In this paper we study the existence of global solutions to the Cauchy problem of nonlinear Schrodinger equation by establishing time weight function spaces and using the contraction mapping principle.
New Exact Solutions for (1 + 1)-Dimensional Dispersion-Less System
Institute of Scientific and Technical Information of China (English)
WANG Xian-Jun; Naranmandula; FU Jing-Li; HU Jian-Guo; BAO Gang; Tubuxin
2008-01-01
Using improved homogeneous balance method, we obtain complex function form new exact solutions for the (1 + l)-dimensional dispersion-less system, and from the exact solutions we derive real function form solution of the field u. Based on this real function form solution, we find some new interesting coherent structures by selecting arbitrary functions appropriately.
Symmetries, Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations
Liu, Han-Ze; Xin, Xiang-Peng
2016-08-01
This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method. Supported by the National Natural Science Foundation of China under Grant Nos. 11171041 and 11505090, Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009, and the doctorial foundation of Liaocheng University under Grant No. 31805
Heuristic and exact solutions to the inverse power index problem for small voting bodies
Kurz, Sascha
2012-01-01
Power indices are mappings that quantify the influence of the members of a voting body on collective decisions a priori. Their nonlinearity and discontinuity makes it difficult to compute inverse images, i.e., to determine a voting system which induces a power distribution as close as possible to a desired one. This paper considers approximations and exact solutions to this inverse problem for the Penrose-Banzhaf index, which are obtained by enumeration and integer linear programming techniques. They are compared to the results of three simple solution heuristics. The heuristics perform well in absolute terms but can be improved upon very considerably in relative terms. The findings complement known asymptotic results for large voting bodies and may improve termination criteria for local search algorithms.
Exact Solutions for Konopelchenko-Dubrovsky Equations%Konopelchenko-Dubrovsky方程的精确解
Institute of Scientific and Technical Information of China (English)
张亚敏
2012-01-01
利用改进的tanh函数展开法,结合Maple环境中Epsilon软件包,求解Konopelchenko-Dubrovsky方程,获得方程若干精确解,同时也体现出改进的tanh函数展开法是一种行之有效的方法,可以广泛的应用于求非线性偏微分方程的精确解.%The extended tanh-function method was applied to study new solutions for Konopelchenko-Dubrovsky equations by means of Epsilon package in Maple. Exact solutions were obtained, Thus the extended tanh-function method is reliable and effective which presents a wider applicability for handling nonlinear partial differential e- quations.
Symmetric Teleparallel Gravity: Some Exact Solutions and Spinor Couplings
Adak, Muzaffer; Sert, Özcan; Kalay, Mestan; Sari, Murat
2013-12-01
In this paper, we elaborate on the symmetric teleparallel gravity (STPG) written in a non-Riemannian space-time with nonzero nonmetricity, but zero torsion and zero curvature. First, we give a prescription for obtaining the nonmetricity from the metric in a peculiar gauge. Then, we state that under a novel prescription of parallel transportation of a tangent vector in this non-Riemannian geometry, the autoparallel curves coincide with those of the Riemannian space-times. Subsequently, we represent the symmetric teleparallel theory of gravity by the most general quadratic and parity conserving Lagrangian with lagrange multipliers for vanishing torsion and curvature. We show that our Lagrangian is equivalent to the Einstein-Hilbert Lagrangian for certain values of coupling coefficients. Thus, we arrive at calculating the field equations via independent variations. Then, we obtain in turn conformal, spherically symmetric static, cosmological and pp-wave solutions exactly. Finally, we discuss a minimal coupling of a spin-1/2 field to STPG.
Some exact solutions of magnetized viscous model in string cosmology
Indian Academy of Sciences (India)
C P Singh
2014-07-01
In this paper, we study anisotropic Bianchi-V Universe with magnetic field and bulk viscous fluid in string cosmology. Exact solutions of the field equations are obtained by using the equation of state (EoS) for a cloud of strings, and a relationship between bulk viscous coefficient and scalar expansion. The bulk viscous coefficient is assumed to be inversely proportional to the expansion scalar. It is interesting to examine the effects of magnetized bulk viscous string model in early and late stages of evolution of the Universe. This paper presents different string models like geometrical (Nambu string), Takabayasi (p-string) and Reddy string models by taking certain physical conditions. We discuss the nature of classical potential for viscous fluid with and without magnetic field. The presence of bulk viscosity stops the Universe from becoming empty in its future evolution. It is observed that the Universe expands with decelerated rate in the presence of viscous fluid with magnetic field whereas, it expands with marginal inflation in the presence of viscous fluid without magnetic field. The other physical and geometrical aspects of each string model are discussed in detail.
Exact Integral Solutions for Two-Phase Flow
McWhorter, David B.; Sunada, Daniel K.
1990-03-01
Exact integral solutions for the horizontal, unsteady flow of two viscous, incompressible fluids are derived. Both one-dimensional and radial displacements are calculated with full consideration of capillary drive and for arbitrary capillary-hydraulic properties. One-dimensional, unidirectional displacement of a nonwetting phase is shown to occur increasingly like a shock front as the pore-size distribution becomes wider. This is in contrast to the situation when an inviscid nonwetting phase is displaced. The penetration of a nonwetting phase into porous media otherwise saturated by a wetting phase occurs in narrow, elongate distributions. Such distributions result in rapid and extensive penetration by the nonwetting phase. The process is remarkably sensitive to the capillary-hydraulic properties that determine the value of knw/kw at large wetting phase saturations, a region in which laboratory measurements provide the least resolution. The penetration of a nonwetting phase can be expected to be dramatically affected by the presence of fissures, worm holes, or other macropores. Calculations for radial displacement of a nonwetting phase resident at a small initial saturation show the displacement to be inefficient. The fractional flow of the nonwetting phase falls rapidly and, for a specific example, becomes 1% by the time one pore volume of water has been injected.
Grooms, Ian
2014-01-01
The non-hydrostatic, quasigeostrophic approximation for rapidly rotating Rayleigh-B\\'enard convection admits a class of exact `single mode' solutions. These solutions correspond to steady laminar convection with a separable structure consisting of a horizontal planform characterized by a single wavenumber multiplied by a vertical amplitude profile, with the latter given as the solution of a nonlinear boundary value problem. The heat transport associated with these solutions is studied in the regime of strong thermal forcing (large reduced Rayleigh number $\\widetilde{Ra}$). It is shown that the Nusselt number $Nu$, a nondimensional measure of the efficiency of heat transport by convection, for this class of solutions is bounded below by $Nu\\gtrsim \\widetilde{Ra}^{3/2}$, independent of the Prandtl number, in the limit of large reduced Rayleigh number. Matching upper bounds include only logarithmic corrections, showing the accuracy of the estimate. Numerical solutions of the nonlinear boundary value problem for ...
Directory of Open Access Journals (Sweden)
Kamruzzaman Khan
2014-04-01
Full Text Available Exact solutions of nonlinear evolution equations (NLEEs play a vital role to reveal the internal mechanism of complex physical phenomena. In this article, we implemented the modified simple equation (MSE method for finding the exact solutions of NLEEs via the (2+1-dimensional cubic Klein–Gordon (cKG equation and the (3+1-dimensional Zakharov–Kuznetsov (ZK equation and achieve exact solutions involving parameters. When the parameters are assigned special values, solitary wave solutions are originated from the exact solutions. It is established that the MSE method offers a further influential mathematical tool for constructing exact solutions of NLEEs in mathematical physics.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Conventional models for fluid flow in well tests have not been consistent with material balance. According to the slightly compressible fluid assumption, the quadratic gradient term in the nonlinear partial differential equation has been usually neglected. This approach is questionable for live oil and low permeability reservoirs. We have already known that linearization by neglecting quadratic gradient terms may lead to errors for large values of well-test time. In this paper, a method that is consistent with material balance was proposed on the spherical flow system. All terms in the nonlinear partial eqiation were retained. Exact solution for the resulting nonlinear partial differential equation in an infinite reservoir was obtained by using the Laplace transform considering wellbore storage. Analytical solution for nonlinear partial differential equation are resulted by using orthogonal transforms under both closed and constant-pressure outer boundary conditions. The law of pressure changes for a fluid compressibility α and a storage coefficient CD were discussed.
Solution of (3+1-Dimensional Nonlinear Cubic Schrodinger Equation by Differential Transform Method
Directory of Open Access Journals (Sweden)
Hassan A. Zedan
2012-01-01
Full Text Available Four-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time. Moreover, as an application of four-dimensional differential transform, exact solutions of nonlinear system of partial differential equations have been investigated. The results of the present method are compared very well with analytical solution of the system. Differential transform method can easily be applied to linear or nonlinear problems and reduces the size of computational work. With this method, exact solutions may be obtained without any need of cumbersome work, and it is a useful tool for analytical and numerical solutions.
Analytical Solution of Nonlinear Problems in Classical Dynamics by Means of Lagrange-Ham
DEFF Research Database (Denmark)
Kimiaeifar, Amin; Mahdavi, S. H; Rabbani, A.
2011-01-01
In this work, a powerful analytical method, called Homotopy Analysis Methods (HAM) is coupled with Lagrange method to obtain the exact solution for nonlinear problems in classic dynamics. In this work, the governing equations are obtained by using Lagrange method, and then the nonlinear governing...
A (1 + 2-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II
Directory of Open Access Journals (Sweden)
Roman Cherniha
2017-01-01
Full Text Available A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1-dimensional. Exact solutions of some (1 + 1-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found.
Energy Technology Data Exchange (ETDEWEB)
Alvarez-Estrada, R.F.
1979-08-01
A comprehensive review of the inverse scattering solution of certain non-linear evolution equations of physical interest in one space dimension is presented. We explain in some detail the interrelated techniques which allow to linearize exactly the following equations: (1) the Korteweg and de Vries equation; (2) the non-linear Schrodinger equation; (3) the modified Korteweg and de Vries equation; (4) the Sine-Gordon equation. We concentrate in discussing the pairs of linear operators which accomplish such an exact linearization and the solution of the associated initial value problem. The application of the method to other non-linear evolution equations is reviewed very briefly.
BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS TO A COUPLED NONLINEAR WAVE SYSTEM
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations, the existence of solitary wave solutions, kink wave solutions, anti-kink wave solutions and periodic wave solutions is obtained. Under different parametric conditions, various suffcient conditions to guarantee the existence of the above so-lutions are given. Some exact explicit parametric representations of travelling wave solutions are derived.
Analytic solutions of nonlinear Cournot duopoly game
Directory of Open Access Journals (Sweden)
Akio Matsumoto
2005-01-01
Full Text Available We construct a Cournot duopoly model with production externality in which reaction functions are unimodal. We consider the case of a Cournot model which has a stable equilibrium point. Then we show the existence of analytic solutions of the model. Moreover, we seek general solutions of the model in the form of nonlinear second-order difference equation.
Indian Academy of Sciences (India)
Zaiyun Zhang; Jianhua Huang; Juan Zhong; Sha-Sha Dou; Jiao Liu; Dan Peng; Ting Gao
2014-06-01
In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (′/)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.
Energy Technology Data Exchange (ETDEWEB)
Williams, Todd O [Los Alamos National Laboratory
2009-01-01
The exact solution for the history-dependent behavior of laminated plates subjected to cylindrical bending is presented. The solution represents the extension of Pagano's solution to consider arbitrary types of constitutive behaviors for the individual lamina as well as arbitrary types of cohesive zones models for delamination behavior. Examples of the possible types of material behavior are plasticity, viscoelasticity, viscoplasticity, and damaging. Examples of possible CZMs that can be considered are linear, nonlinear hardening, as well as nonlinear with softening. The resulting solution is intended as a benchmark solution for considering the predictive capabilities of different plate theories. Initial results are presented for several types of history-dependent material behaviors. It is shown that the plate response in the presence of history-dependent behaviors can differ dramatically from the elastic response. These results have strong implications for what constitutes an appropriate plate theory for modeling such behaviors.
An Exact Solution to the Two-Particle Boltzmann Equation System for Maxwell Gases
Institute of Scientific and Technical Information of China (English)
布仁满都拉; 赵迎春
2012-01-01
An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.
Numerical and Exact Solution of Buckling Load For Beam on Elastic Foundation
Directory of Open Access Journals (Sweden)
Roland JANČO
2013-06-01
Full Text Available In this paper we will be presented the exact solution of buckling load for supported beam on elastic foundation. Exact solution will be compared with numerical solution by FEM in our code in Matlab. Implementation of buckling to FEM will be presented here.
Dynamical understanding of loop soliton solution for several nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
Ji-bin LI
2007-01-01
It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.
Institute of Scientific and Technical Information of China (English)
ZHANG Shan-Qing; LI Zhi-Bin
2004-01-01
@@ The master equation of a one-dimensional lattice-gas model with order preservation where the occupation probabilities of sites corresponding to Bose statistics as a consequence of the prescribed dynamics is studied with the potential symmetry method. The infinite-parameter potential symmetry and a new exact solution are obtained. The result illustrates that there remains the possibility of the above nonlinear equation to a linear partial differential equation by a non-invertible mapping.
Indian Academy of Sciences (India)
BHARDWAJ S B; SINGH RAM MEHAR; SHARMA KUSHAL; MISHRA S C
2016-06-01
Attempts have been made to explore the exact periodic and solitary wave solutions of nonlinear reaction diffusion (RD) equation involving cubic–quintic nonlinearity along with timedependent convection coefficients. Effect of varying model coefficients on the physical parameters of solitary wave solutions is demonstrated. Depending upon the parametric condition, the periodic,double-kink, bell and antikink-type solutions for cubic–quintic nonlinear reaction-diffusion equation are extracted. Such solutions can be used to explain various biological and physical phenomena.
A Series of Exact Solutions of (2+1)-Dimensional CDGKS Equation
Institute of Scientific and Technical Information of China (English)
YANG Zong-Hang
2006-01-01
An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guideline to classify the various types of the solution according to some parameters.
Coughlin, Eric R
2016-01-01
We present the exact solutions for the collapse of a spherically-symmetric, cold (i.e., pressureless) cloud under its own self-gravity, valid for arbitrary initial density profiles and not restricted to the realm of self-similarity. These solutions exhibit a number of remarkable features, including the self-consistent formation of and subsequent accretion onto a central point mass. A number of specific examples are provided, and we show that Penston's solution of pressureless, self-similar collapse is recovered for polytropic density profiles; importantly, however, we demonstrate that the time over which this solution holds is fleetingly narrow, implying that much of the collapse proceeds non-self-similarly. We show that our solutions can naturally incorporate turbulent pressure support, and we investigate the evolution of overdensities -- potentially generated by such turbulence -- as the collapse proceeds. Finally, we analyze the evolution of the angular velocity and magnetic fields in the limit that their ...
Directory of Open Access Journals (Sweden)
Huanhe Dong
2014-01-01
Full Text Available We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalized Dp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value of p is 5.
Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations
Directory of Open Access Journals (Sweden)
Zeid I. A. Al-Muhiameed
2012-01-01
Full Text Available With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parameters p and q in the obtained solutions by using the computer simulation.
Nonholonomic jet deformations, exact solutions for modified Ricci soliton and Einstein equations
Rajpoot, Subhash; Vacaru, Sergiu I.
Let g be a pseudo-Riemannian metric of arbitrary signature on a manifold V with conventional n + n-dimensional splitting, n ≥ 2, determined by a nonholonomic (nonintegrable) distribution 𝒩 defining a generalized (nonlinear) connection and associated nonholonomic frame structures. We work with an adapted linear metric compatible connection D ̂ and its nonzero torsion 𝒯 ̂, both completely determined by g. Our first goal is to prove that there are certain generalized frame and/or jet transforms and prolongations with (g,V) → (g ̂,V ̂) into explicit classes of solutions of some generalized Einstein equations R ̂ic = Λg ̂, Λ = const, encoding various types of (nonholonomic) Ricci soliton configurations and/or jet variables and symmetries. The second goal is to solve additional constraint equations for zero torsion, 𝒯 ̂ = 0, on generalized solutions constructed in explicit forms with jet variables and extract Levi-Civita configurations. This allows us to find generic off-diagonal exact solutions depending on all space time coordinates on V via generating and integration functions and various classes of constant jet parameters and associated symmetries. Our third goal is to study how such generalized metrics and connections can be related by the so-called “half-conformal” and/or jet deformations of certain subclasses of solutions with one, or two, Killing symmetries. Finally, we present some examples of exact solutions constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by nonholonomic jet structures and generalized connections.
Directory of Open Access Journals (Sweden)
Shaheed N. Huseen
2013-01-01
Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
Explicit solutions of nonlinear wave equation systems
Institute of Scientific and Technical Information of China (English)
Ahmet Bekir; Burcu Ayhan; M.Naci (O)zer
2013-01-01
We apply the (G'/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions,trigonometric functions,and rational functions with arbitrary parameters.We highlight the power of the (G'/G)-expansion method in providing generalized solitary wave solutions of different physical structures.It is shown that the (G'/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.
Exact solution of the relativistic quantum Toda chain
Zhang, Xin; Yang, Wen-Li; Shi, Kangjie; Wang, Yupeng
2016-01-01
The relativistic quantum Toda chain model is studied with the generalized algebraic Bethe Ansatz method. By employing a set of local gauge transformations, proper local vacuum states can be obtained for this model. The exact spectrum and eigenstates of the model are thus constructed simultaneously.
Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2+1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.
Travelling wave solutions to nonlinear physical models by means of the ﬁrst integral method
Indian Academy of Sciences (India)
İsmail Aslan Aslan
2011-04-01
This paper presents the ﬁrst integral method to carry out the integration of nonlinear partial differential equations in terms of travelling wave solutions. For illustration, three important equations of mathematical physics are analytically investigated. Through the established ﬁrst integrals, exact solutions are successfully constructed for the equations considered.
Fully nonlinear and exact perturbations of the Friedmann world model: non-flat background
Energy Technology Data Exchange (ETDEWEB)
Noh, Hyerim, E-mail: hr@kasi.ac.kr [Korea Astronomy and Space Science Institute, Daejeon, 305-348 (Korea, Republic of)
2014-07-01
We extend the fully non-linear and exact cosmological perturbation equations in a Friedmann background universe to include the background curvature. The perturbation equations are presented in a gauge ready form, so any temporal gauge condition can be adopted freely depending on the problem to be solved. We consider the scalar, and vector perturbations without anisotropic stress. As an application, we analyze the equations in the special case of irrotational zero-pressure fluid in the comoving gauge condition. We also present the fully nonlinear formulation for a minimally coupled scalar field.
Fully nonlinear and exact perturbations of the Friedmann world model: Non-flat background
Noh, Hyerim
2014-01-01
We extend the fully non-linear and exact cosmological perturbation equations in a Friedmann background universe to include the background curvature. The perturbation equations are presented in a gauge ready form, so any temporal gauge condition can be adopted freely depending on the problem to be solved. %The background curvature term explicitly appears only in the energy and momentum constraint equations. We consider the scalar, and vector perturbations without anisotropic stress. As an application, we analyze the equations in the special case of irrotational zero-pressure fluid in the comoving gauge condition. We also present the fully nonlinear formulation for a minimally coupled scalar field.
Denicol, Gabriel S; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael
2014-01-01
We present an exact solution to the Boltzmann equation which describes a system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion for arbitrary shear viscosity to entropy density ratio. This new solution is constructed by considering the conformal map between Minkowski space and the direct product of three dimensional de Sitter space with a line. The resulting solution respects SO(3)_q x SO(1,1) x Z_2 symmetry. We compare the exact kinetic solution with exact solutions of the corresponding macroscopic equations with the same symmetry that were obtained from the kinetic theory in ideal and second-order viscous hydrodynamic approximations.
An Exact Formal Solution to Reaction-Diffusion Equations from Biomathematics
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
We study the exact formal solution to the simplified Keller-Segel system modelling chemotaxis. The method we use is series expanding. The main result is to attain the formal solution to the simplified Keller-Segel system.
Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach
Panahi, H.; Baradaran, M.; Savadi, A.
2015-10-01
Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.
Generalized nonlinear Proca equation and its free-particle solutions
Energy Technology Data Exchange (ETDEWEB)
Nobre, F.D. [Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, RJ (Brazil); Plastino, A.R. [Universidad Nacional Buenos Aires-Noreoeste, CeBio y Secretaria de Investigacion, Junin (Argentina)
2016-06-15
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ{sup μ}(vector x,t), involves an additional field Φ{sup μ}(vector x,t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E{sup 2} = p{sup 2}c{sup 2} + m{sup 2}c{sup 4} for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed. (orig.)
New Exact Solutions of the Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces
Institute of Scientific and Technical Information of China (English)
LI De-Sheng; ZHANG Hong-Qing
2004-01-01
In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.
On Exact Solutions to Partial Differential Equations by the Modified Homotopy Perturbation Method
Institute of Scientific and Technical Information of China (English)
Gang YANG; Ru-yun CHEN; Luo-gen YAO
2012-01-01
Based on the modified homotopy perturbation method (MHPM),exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions.Under suitable initial conditions,the PDE is transformed into an ODE.Some illustrative examples reveal the efficiency of the proposed method.
Periodic solutions of nonlinear vibrating beams
Directory of Open Access Journals (Sweden)
J. Berkovits
2003-01-01
Full Text Available The aim of this paper is to prove new existence and multiplicity results for periodic semilinear beam equation with a nonlinear time-independent perturbation in case the period is not prescribed. Since the spectrum of the linear part varies with the period, the solvability of the equation depends crucially on the period which can be chosen as a free parameter. Since the period of the external forcing is generally unknown a priori, we consider the following natural problem. For a given time-independent nonlinearity, find periods T for which the equation is solvable for any T-periodic forcing. We will also deal with the existence of multiple solutions when the nonlinearity interacts with the spectrum of the linear part. We show that under certain conditions multiple solutions do exist for any small forcing term with suitable period T. The results are obtained via generalized Leray-Schauder degree and reductions to invariant subspaces.
The approximate solutions of nonlinear Boussinesq equation
Lu, Dianhen; Shen, Jie; Cheng, Yueling
2016-04-01
The homotopy analysis method (HAM) is introduced to solve the generalized Boussinesq equation. In this work, we establish the new analytical solution of the exponential function form. Applying the homotopy perturbation method to solve the variable coefficient Boussinesq equation. The results indicate that this method is efficient for the nonlinear models with variable coefficients.
Energy Technology Data Exchange (ETDEWEB)
Kong Cuicui [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)], E-mail: cuicuikong@yahoo.com.cn; Wang Dan; Song Lina; Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2009-01-30
In this paper, with the aid of symbolic computation and a general ansaetz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansaetz. The method can also be applied to other nonlinear partial differential equations.
Exact Numerical Solutions of Bose-Hubbard Model
Institute of Scientific and Technical Information of China (English)
ZHANG Dan; PAN Feng
2004-01-01
Hamiltonian of a one-dimensional Bose-Hubbard model is re-formulated by using differential realization of the boson algebra. Energy matrices can then be generated systematically by using a Mathematica package. The output can be taken as the input of other diagonalization codes. As examples, exact energy eigenvalues and the corresponding wavefunctions for some cases are obtained with a Fortran diagonalization code. Phase transition of the model is analyzed.
Exact solution of an su(n) spin torus
Hao, Kun; Li, Guang-Liang; Yang, Wen-Li; Shi, Kangjie; Wang, Yupeng
2016-01-01
The trigonometric su(n) spin chain with anti-periodic boundary condition (su(n) spin torus) is demonstrated to be Yang-Baxter integrable. Based on some intrinsic properties of the R-matrix, certain operator product identities of the transfer matrix are derived. These identities and the asymptotic behavior of the transfer matrix together allow us to obtain the exact eigenvalues in terms of an inhomogeneous T-Q relation via the off-diagonal Bethe Ansatz.
Exact solution of an su(n) spin torus
Hao, Kun; Cao, Junpeng; Li, Guang-Liang; Yang, Wen-Li; Shi, Kangjie; Wang, Yupeng
2016-07-01
The trigonometric su(n) spin chain with anti-periodic boundary condition (su(n) spin torus) is demonstrated to be Yang-Baxter integrable. Based on some intrinsic properties of the R-matrix, certain operator product identities of the transfer matrix are derived. These identities and the asymptotic behavior of the transfer matrix together allow us to obtain the exact eigenvalues in terms of an inhomogeneous T - Q relation via the off-diagonal Bethe Ansatz.
Multi-soliton rational solutions for some nonlinear evolution equations
Directory of Open Access Journals (Sweden)
Osman Mohamed S.
2016-01-01
Full Text Available The Korteweg-de Vries equation (KdV and the (2+ 1-dimensional Nizhnik-Novikov-Veselov system (NNV are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially integrable equations. Compared with Hirota’s method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.
The hyperbolic Allen-Cahn equation: exact solutions
Nizovtseva, I. G.; Galenko, P. K.; Alexandrov, D. V.
2016-10-01
Using the first integral method, a general set of analytical solutions is obtained for the hyperbolic Allen-Cahn equation. The solutions are presented by (i) the class of continual solutions described by \\tanh -profiles for traveling waves of the order parameter, and (ii) the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system. It is shown that the solutions include the previous analytical results for the parabolic Allen-Cahn equation as a limited class of \\tanh -functions, in which the inertial effects are omitted.
Observational Constraints on New Exact Inflationary Scalar-field Solutions
Barrow, John D
2016-01-01
An algorithm is used to generate new solutions of the scalar field equations in homogeneous and isotropic universes. Solutions can be found for pure scalar fields with various potentials in the absence and presence of spatial curvature and other perfect fluids. A series of generalisations of the Chaplygin gas and bulk viscous cosmological solutions for inflationary universes are found. We also show how the Hubble slow-roll parameters can be calculated using the solution algorithm and we compare these inflationary solutions with the observational data provided by the Planck 2015 collaboration in order to constraint and rule out some of these models.
Nonlinear Helicons ---an analytical solution elucidating multi-scale structure
Abdelhamid, Hamdi M
2016-01-01
The helicon waves exhibit varying characters depending on plasma parameters, geometry, and wave numbers. Here we elucidate an intrinsic multi-scale property embodied by the combination of dispersive effect and nonlinearity. The extended magnetohydrodynamics model (exMHD) is capable of describing wide range of parameter space. By using the underlying Hamiltonian structure of exMHD, we construct an exact nonlinear solution which turns out to be a combination of two distinct modes, the helicon and Trivelpiece-Gould (TG) waves. In the regime of relatively low frequency or high density, however, the combination is made of the TG mode and an ion cyclotron wave (slow wave). The energy partition between these modes is determined by the helicities carried by the wave fields.
Gupta, A. K.; Ray, S. Saha
2014-09-01
In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK) equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.
Directory of Open Access Journals (Sweden)
A. K. Gupta
2014-09-01
Full Text Available In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.
Application of differential constraint method to exact solution of second-grade fluid
Institute of Scientific and Technical Information of China (English)
Dao-xiang ZHANG; Su-xiao FENG; Zhi-ming LU; Yu-lu LIU
2009-01-01
A differential constraint method is used to obtain analytical solutions of a second-grade fluid flow. By using the first-order differential constraint condition, exact solutions of Poiseuille flows, jet flows and Couette flows subjected to suction or blowing forces, and planar elongational flows are derived. In addition, two new classes of exact solutions for a second-grade fluid flow are found. The obtained exact solutions show that the non-Newtonian second-grade flow behavior depends not only on the material viscosity but also on the material elasticity. Finally, some boundary value problems are discussed.
Analytical solution of strongly nonlinear Duffing oscillators
Directory of Open Access Journals (Sweden)
A.M. El-Naggar
2016-06-01
Full Text Available In this paper, a new perturbation technique is employed to solve strongly nonlinear Duffing oscillators, in which a new parameter α=α(ε is defined such that the value of α is always small regardless of the magnitude of the original parameter ε. Therefore, the strongly nonlinear Duffing oscillators with large parameter ε are transformed into a small parameter system with respect to α. Approximate solution obtained by the present method is compared with the solution of energy balance method, homotopy perturbation method, global error minimization method and lastly numerical solution. We observe from the results that this method is very simple, easy to apply, and gives a very good accuracy not only for small parameter εbut also for large values of ε.
Solutions manual to accompany Nonlinear programming
Bazaraa, Mokhtar S; Shetty, C M
2014-01-01
As the Solutions Manual, this book is meant to accompany the main title, Nonlinear Programming: Theory and Algorithms, Third Edition. This book presents recent developments of key topics in nonlinear programming (NLP) using a logical and self-contained format. The volume is divided into three sections: convex analysis, optimality conditions, and dual computational techniques. Precise statements of algortihms are given along with convergence analysis. Each chapter contains detailed numerical examples, graphical illustrations, and numerous exercises to aid readers in understanding the concepts a
Searching Exact Solutions for Compact Stars in Braneworld: a conjecture
2007-01-01
In the context of the braneworld, a method to find consistent solutions to Einstein's field equations in the interior of a spherically symmetric, static and non uniform stellar distribution with Weyl stresses is developed. This method, based in the fact that any braneworld stellar solution must have the general relativity solution as a limit, produces a constraint which reduces the degrees of freedom on the brane. Hence the non locality and non closure of the braneworld equations can be overc...
Exact anisotropic viscous fluid solutions of Einstein's equations
Goenner, H. F. M.; Kowalewski, F.
1989-05-01
A method for obtaining anisotropic, rotationless viscous fluid matter solutions of Bianchi type I and Segré type [1, 111] with the barotropic equation of state is presented. Solutions for which the anisotropy decreases exponentially or with a power law as well as solutions with average Hubble parameterH ˜t -1 are discussed. Also, a class of solutions with constant anisotropy and Bianchi type VIh is found. The dominant energy condition holds and the transport coefficients show the right sign.
Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas
Daghan, D.; Donmez, O.
2016-06-01
Traveling wave solution of the Gardner equation is studied analytically by using the two dependent ( G '/ G,1/ G)-expansion and (1/ G ')-expansion methods and direct integration. The exact solutions of the Gardner equations are obtained. Our analytic solutions are applied to the unmagnetized four-component and dusty plasma systems consisting of hot protons and electrons to investigate dynamical features of the solitons and shock waves produced in these systems. A wide variety of parameters of the plasma is used, and the basic features of the Gardner solitons that are beyond the existing study in literature are found. It is observed that the analytic solutions from ( G '/ G,1/ G)-expansion and (1/ G ')-expansion methods only produce shock waves but the solitary waves are found from the analytic solutions derived from the direct integration. It is also noted that the superhot electrons and relative mass density of the electrons significantly effect the soliton's amplitude, width, and position. We have also numerically proved that the combination of every value of nomalized density μ 1 or temperature ratio σ 1 with the other sets of plasma parameters creates a region where the solutions have similar physical properties. The time-dependent behavior of the soliton is also studied, and a periodic motion of soliton along the phase variable η is found during the evolution. The investigations and the limits presented in this study may be helpful for studying and understanding the nonlinear properties of the solitary and shock waves seen in various physical and astrophysical plasma systems.
Exact solution of quantum gravity in 1 + 1 dimensions
Rajeev, S. G.
1982-06-01
Quantum gravity in 1 + 1 dimensions, with zero cosmological constant is formulated, including contributions from all possible topologies. The spectrum and the S-matrix are calculated exactly. Pure gravity is found to exist in a disordered phase, dominated by topologically non-trivial configurations. In the presence of fermionic matter fields, space-time can undergo a phase transition to an ordered phase. It is a pleasure to thank Professor A.P. Balachandran, Professor R. Shankar and Professor A. Ashtekar, and C.G. Trahern, V.P. Nair and V. Rodgers.
Exact solutions for the 2d one component plasma
Andersen, Timothy D
2011-01-01
The 2d one component gas of pointlike charges in a uniform neutralizing background interacting with a logarithmic potential is a common model for plasmas. In its classical equilibrium statistics at fixed temperature (canonical ensemble) it is formally related to certain types of random matrices with Gaussian distribution and complex eigenvalues. In this paper, I present an exact integration of this ensemble for $N$ such particles (or alternatively $N\\times N$ matrices) for all complex temperatures, a significant open problem in statistical physics for several decades.
Exact Solutions of the Photon Equation in Anisotropic Spacetimes
Havare, A; Aydogdu, O; Salti, M; Yetkin, T; Havare, Ali; Korunur, Murat; Aydogdu, Oktay; Salti, Mustafa; Yetkin, Taylan
2005-01-01
In this paper we study solution of the photon equation (the Massless Duffin-Kemmer-Petiau equation (mDKP)) in anisotropic expanding the Bianchi-I type spacetime using the Fourier analyze method. The harmonic oscillator behavior of the solutions is found. It is shown that Maxwell equations are equivalent to the photon equation.
New Families of Exact Solutions for (2+1)-Dimensional Broer-Kaup System
Institute of Scientific and Technical Information of China (English)
ZHAO Hong; BAI Cheng-Lin; HAN Ji-Guang
2005-01-01
Using a further modified extended tanh-function method, rich new families of the exact solutions for the (2+ 1)-dimensional Broer-Kaup (BK) system, comprising the non-traveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained.
Exact Shock Solution of a Coupled System of Delay Differential Equations: A Car-Following Model
Tutiya, Yohei; Kanai, Masahiro
2007-08-01
In this letter, we present exact shock solutions of a coupled system of delay differential equations, which was introduced as a traffic-flow model called car-following model. We use the Hirota method, originally developed in order to solve soliton equations. The relevant delay differential equations have been known to allow exact solutions expressed by elliptic functions with periodic boundary conditions. In the present work, however, shock solutions are obtained with open boundaries, representing the stationary propagation of a traffic jam.
Testing the Cactus code on exact solutions of Einstein field equations
Vulcanov, D.; M. Alcubierre
2002-01-01
The article presents a series of numerical simulations of exact solutions of the Einstein equations performed using the Cactus code, a complete 3-dimensional machinery for numerical relativity. We describe an application (''''thorn'') for the Cactus code that can be used for evolving a variety of exact solutions, with and without matter, including solutions used in modern cosmology for modeling the early stages of the universe. Our main purpose has been to test the Cactus code on these well-k...
Directory of Open Access Journals (Sweden)
Taha Aziz
2013-01-01
Full Text Available The simplest equation method is employed to construct some new exact closed-form solutions of the general Prandtl's boundary layer equation for two-dimensional flow with vanishing or uniform mainstream velocity. We obtain solutions for the case when the simplest equation is the Bernoulli equation or the Riccati equation. Prandtl's boundary layer equation arises in the study of various physical models of fluid dynamics. Thus finding the exact solutions of this equation is of great importance and interest.
An exactly solvable model of an oscillator with nonlinear coupling and zeros of Bessel functions
Dodonov, V. V.; Klimov, A. B.
1993-01-01
We consider an oscillator model with nonpolynomial interaction. The model admits exact solutions for two situations: for energy eigenvalues in terms of zeros of Bessel functions, that were considered as functions of the continuous index; and for the corresponding eigenstates in terms of Lommel polynomials.
EXACT LINEARIZATION BASED MULTIPLE-SUBSPACE ITERATIVE RESOLUTION TO AFFINE NONLINEAR CONTROL SYSTEM
Institute of Scientific and Technical Information of China (English)
XU Zi-xiang; ZHOU De-yun; DENG Zi-chen
2006-01-01
To the optimal control problem of affine nonlinear system, based on differential geometry theory, feedback precise linearization was used. Then starting from the simulative relationship between computational structural mechanics and optimal control,multiple-substructure method was inducted to solve the optimal control problem which was linearized. And finally the solution to the original nonlinear system was found. Compared with the classical linearizational method of Taylor expansion, this one diminishes the abuse of error expansion with the enlargement of used region.
Exact time-localized solutions in Vacuum String Field Theory
Bonora, L; Santos, R J S; Tolla, D D
2004-01-01
We address the problem of finding star algebra projectors that exhibit localized time profiles. We use the double Wick rotation method, starting from an Euclidean (unconventional) lump solution, which is characterized by the Neumann matrix being the conventional one for the continuous spectrum, while the inverse of the conventional one for the discrete spectrum. This is still a solution of the projector equation and we show that, after inverse Wick-rotation, its time profile has the desired localized time dependence. We study it in detail in the low energy regime (field theory limit) and in the extreme high energy regime (tensionless limit) and show its similarities with the rolling tachyon solution.
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
5D Lovelock gravity: new exact solutions with torsion
Cvetković, B
2016-01-01
We investigate Lovelock gravity in five dimensions in first order formalism. We construct a new class of solutions: BTZ black ring with(out) torsion. We show that our solution with torsion exists in the different sector of the Lovelock gravity compared to Lovelock Chern-Simons and the sector investigated by Canfora et al. We compute conserved charges of the solutions by using Nester's formula, and confirm the results by canonical method. We show that the theory linearized around the background with torsion possesses additional degrees of freedom compared to general relativity.
Exact half-BPS type IIB interface solutions I: local solution and supersymmetric Janus
Energy Technology Data Exchange (ETDEWEB)
D' Hoker, Eric; Estes, John; Gutperle, Michael [Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 (United States)
2007-06-15
The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold AdS{sub 4} x S{sup 2} x S{sup 2} x {sigma} with SO(2, 3) x SO(3) x SO(3) symmetry in terms of two holomorphic functions on a Riemann surface {sigma}, which generally has a boundary. This is achieved by reducing the BPS equations using the above symmetry requirements, proving that all solutions of the BPS equations solve the full Type IIB supergravity field equations, mapping the BPS equations onto a new integrable system akin to the Liouville and Sine-Gordon theories, and mapping this integrable system to a linear equation which can be solved exactly. Amongst the infinite class of solutions, a non-singular Janus solution is identified which provides the AdS/CFT dual of the maximally supersymmetric Yang-Mills interface theory discovered recently. The construction of general classes of globally non-singular solutions, including fully back-reacted AdS{sub 5} x S{sup 5} and supersymmetric Janus doped with D5 and/or NS5 branes, is deferred to a companion paper.
Charged-Surface Instability Development in Liquid Helium; Exact Solutions
Zubarev, N M
2000-01-01
The nonlinear dynamics of charged-surface instability development was investigated for liquid helium far above the critical point. It is found that, if the surface charge completely screens the field above the surface, the equations of three-dimensional (3D) potential motion of a fluid are reduced to the well-known equations describing the 3D Laplacian growth process. The integrability of these equations in 2D geometry allows the analytic description of the free-surface evolution up to the formation of cuspidal singularities at the surface.
New exact perfect fluid solutions of Einstein's equations. II
Uggla, Claes; Rosquist, Kjell
1990-12-01
A family of new spatially homogeneous Bianchi type VIh perfect fluid solutions of the Einstein equations is presented. The fluid flow is orthogonal to the spatially homogeneous hypersurfaces, and the pressure is proportional to the energy density.
Exact Solutions of Bogoyavlenskii Coupled KdV Equations
Institute of Scientific and Technical Information of China (English)
HUHeng-Chun; LOUSen-Yue
2004-01-01
The special soliton solutions of Bogoyavlenskii coupled KdV equations are obtained by means of the standard Weiss-Tabor -Carnvale Painleve' truncation expansion and the nonstandard truncation of a modified Conte's invariant Painlevé expansion.
EXACT SOLUTIONS FOR MAGNETOHYDRODYNAMIC FLOW IN A ROTATING FLUID
Institute of Scientific and Technical Information of China (English)
S.Asghar; Masood Khan; A.M.Siddiqui; T.Hayat
2002-01-01
An analytical solution is obtained for the flow due to solid-body rotations of an oscillating porous disk and of a fluid at infinity. Neglecting the induced magnetic field, the effects of the transversely applied magnetic field on the flow are studied. Further, the flow confined between two disks is also discussed. It is found that an infinite number of solutions exist for the flow confined between two disks.
Explicit Traveling Wave Solutions to Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
Linghai ZHANG
2011-01-01
First of all,some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations,nonlinear dissipative dispersive wave equations,nonlinear convection equations,nonlinear reaction diffusion equations and nonlinear hyperbolic equations,respectively.
Directory of Open Access Journals (Sweden)
Hassan A. Zedan
2013-01-01
Full Text Available We introduce two powerful methods to solve the Davey-Stewartson equations: one is the homotopy perturbation method (HPM and the other is the homotopy analysis method (HAM. HAM is a strong and easy to use analytic tool for nonlinear problems. Comparison of the HPM results with the HAM results, and compute the absolute errors between the exact solutions of the DS equations with the HPM solutions and HAM solutions are obtained.
Energy Technology Data Exchange (ETDEWEB)
Zhang Sheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China)], E-mail: zhshaeng@yahoo.com.cn
2008-03-10
In this Letter, the Exp-function method is used to seek generalized solitonary solutions of Riccati equation. Based on the Riccati equation and its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional Broer-Kaup-Kupershmidt equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.
Mushy-zone model with an exact solution
Energy Technology Data Exchange (ETDEWEB)
Solomon, A. D.; Wilson, D. G.; Alexiades, V.
1982-04-01
In this paper we propose a very simple model of a mushy zone which admits of an explicit solution. To our knowledge, it is the only instance where an actual observation of the mushy zone width and structure is used as a partial basis for the model definition. The model rests upon two unknown parameters. The first determines the relation between the equilibrium temperature gradient and the mushy zone width. The second depends upon the dendritic structure in the mushy zone, and is related to the solid fraction. Both can be estimated from experiments. We will limit ourselves to defining the model, presenting its closed form solution, and giving tables from which the solution can be found explicitly. It is shown that in most cases the predicted mushy zone is of very negligible importance.
Three-dimensional dilatonic gravity's rainbow: Exact solutions
Hossein Hendi, Seyed; Eslam Panah, Behzad; Panahiyan, Shahram
2016-10-01
Deep relations of dark energy scenario and string theory results into dilaton gravity, on the one hand, and the connection between quantum gravity and gravity's rainbow, on the other hand, motivate us to consider three-dimensional dilatonic black hole solutions in gravity's rainbow. We obtain two classes of the solutions, which are polynomial and logarithmic forms. We also calculate conserved and thermodynamic quantities, and examine the first law of thermodynamics for both classes. In addition, we study thermal stability and show that one of the classes is thermally stable while the other one is unstable.
Three-dimensional dilatonic gravity's rainbow: Exact solutions
Hossein Hendi, Seyed; Eslam Panah, Behzad; Panahiyan, Shahram
2016-10-01
Deep relations of dark energy scenario and string theory results into dilaton gravity, on the one hand, and the connection between quantum gravity and gravity's rainbow, on the other hand, motivate us to consider three-dimensional dilatonic black hole solutions in gravity's rainbow. We obtain two classes of the solutions, which are polynomial and logarithmic forms. We also calculate conserved and thermodynamic quantities, and examine the first law of thermodynamics for both classes. In addition, we study thermal stability and show that one of the classes is thermally stable while the other one is unstable.
Families of exact solutions of a 2D gravity model minimally coupled to electrodynamics
Moayedi, S K
2001-01-01
Three families of exact solutions for 2-dimensional gravity minimally coupled to electrodynamics are obtained in the context of ${\\cal R}=T$ theory. It is shown, by supersymmetric formalism of quantum mechanics, that the quantum dynamics of a neutral bosonic particle on static backgrounds with both varying curvature and electric field is exactly solvable.
Exact solutions of the high dimensional hard-core Fermi-Hubbard model
Institute of Scientific and Technical Information of China (English)
潘峰; 戴连荣
2001-01-01
A simple algebraic approach to exact solutions of the hard-core Fermi-Hubbard model is proposed. Excitation energies and the corresponding wavefunctions of the hard-core Fermi-Hubbard model with nearest neighbor hopping cases in high dimension are obtained by using this method, which manifests that the model is exactly solvable in any dimension.
Some Exact Solutions of the Semilocal Popov Equations with Many Flavors
Kim, Chanju
2015-01-01
In 2+1 dimensional nonrelativistic Chern-Simons gauge theories on $S^2$ which has a global $SU(M)$ symmetry, the semilocal Popov vortex equations are obtained as Bogomolny equations by minimizing the energy in the presence of a uniform external magnetic field. We study the equations with many flavors and find several families of exact solutions. The equations are transformed to the semilocal Liouville equations for which some exact solutions are known. In this paper, we find new exact solutions of the semilocal Liouville equations. Using these solutions, we construct solutions to the semilocal Popov equations. The solutions are expressed in terms of one or more arbitrary rational functions on $S^2$. Some simple solutions reduce to $CP^{M-1}$ lump configurations.
Energy Technology Data Exchange (ETDEWEB)
Xie, Ming [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
2000-05-18
Exact solution and variational approximation of eigenmodes in high gain FELs are presented. These eigenmodes specify transverse profiles and exponential growth rates of the laser field before saturation. They are self-consistent solutions of coupled Maxwell–Vlasov equations describing FEL interaction taking into account the effects due to energy spread, emittance and betatron oscillations of the electron beam, as well as diffraction and optical guiding of the laser field. A new formalism of scaling is introduced and based on which solutions in various limiting cases are discussed. Additionally, a fitting formula is obtained from interpolating the variational solution for quick calculation of exponential growth rate of the fundamental mode.
Exact String-Like Solutions in Conformal Gravity
Verbin, Y
2010-01-01
The Cylindrically-symmetric static vacuum equations of Conformal Gravity are solved for the case of additional boost symmetry along the axis. We present the complete family of solutions which describe the exterior gravitational field of line sources in Conformal Gravity. We also analyze the null geodesics in these spaces.
Exact Jacobian Elliptic Function Solutions to sinh-Gordon Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Kuo; LIU Shi-Da
2006-01-01
In this paper, two transformations are introduced to solve sinh-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions.It is shown that different transformations are required in order to obtain more kinds of solutions to the sinh-Gordon equation.
Institute of Scientific and Technical Information of China (English)
FENG Qing-Hua
2013-01-01
In this paper,an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative.By a fractional complex transformation,a given fractional differential-difference equation can be turned into another differential-difference equation of integer order.The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system.As a result,some new exact solutions including hyperbolic function solutions,trigonometric function solutions and rational solutions are established.
Feng, Qing-Hua
2013-05-01
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.
Institute of Scientific and Technical Information of China (English)
田野; 陈静; 张忠飞
2012-01-01
In this paper, the separation transformation approach is extended to the （N ＋ 1）-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of SHe superfluid. This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation. Then the general solutions of the set of partial differential equations are obta/ned and the nonlinear ordinary differential equation is solved by F-expansion method. Finally, many new exact solutions of the （N ＋ 1）-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation. For the case of N 〉 2, there is an arbitrary function in the exact solutions, which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.
A Large Class of Exact Solutions to the One-Dimensional Schrodinger Equation
Karaoglu, Bekir
2007-01-01
A remarkable property of a large class of functions is exploited to generate exact solutions to the one-dimensional Schrodinger equation. The method is simple and easy to implement. (Contains 1 table and 1 figure.)
Adomian Decomposition Method and Exact Solutions of the Perturbed KdV Equation
Institute of Scientific and Technical Information of China (English)
WU Bin; LOU Sen-Yue
2002-01-01
The Adomian decomposition method is used to solve the Cauchy problem of the perturbed KdV equation.Three types of exact solitary wave solutions are reobtained via the A domian's approach by selecting the initial conditionsappropriately.
Painlevé analysis and exact solutions of a modified Boussinesq equation
Liu, Q P
1995-01-01
We consider a modified Boussinesq type equation. The Painlev\\'{e} test of the WTC method is performed for this equation and it shows that the equation has weak Painlev\\'{e} property. Some exact solutions are constructed.
Adomian Decomposition Method and Exact Solutions of the Perturbed KdV Equation
Institute of Scientific and Technical Information of China (English)
WuBin; LOUSen－Yue
2002-01-01
The Adomian decomposition method is used to solve the Cauchy problem of the perturbed KdV equation.Three types of exact solitary wave solutions are reobtained via the Adomian's approach by selcting the initial conditions appropriately.
Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain
Institute of Scientific and Technical Information of China (English)
LIU Ping; SHI JunPing; WANG YuWen
2008-01-01
We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain.We obtain the precise bifurcation diagrams.
Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain
Institute of Scientific and Technical Information of China (English)
2008-01-01
We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.
Exact solutions of a class of fractional Hamiltonian equations involving Caputo derivatives
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Baleanu, Dumitru [Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara 06530 (Turkey); Trujillo, Juan J [Departamento de Analisis Matematico, University of La Laguna, 38271 La Laguna, Tenerife (Spain)], E-mail: dumitru@cankaya.edu.tr, E-mail: JTrujill@ullmat.es, E-mail: baleanu@venus.nipne.ro
2009-11-15
The fractional Hamiltonian equations corresponding to the Lagrangians of constrained systems within Caputo derivatives are investigated. The fractional phase space is obtained and the exact solutions of some constrained systems are obtained.
Bäcklund Transformation and New Exact Solutions of the Sharma-Tasso-Olver Equation
Directory of Open Access Journals (Sweden)
Lin Jianming
2011-01-01
Full Text Available The Sharma-Tasso-Olver (STO equation is investigated. The Painlevé analysis is efficiently used for analytic study of this equation. The Bäcklund transformations and some new exact solutions are formally derived.
Exact Traveling Wave Solutions for a Kind of Generalized Ginzburg-Landau Equation
Institute of Scientific and Technical Information of China (English)
LIU Cheng-Shi
2005-01-01
Using a complete discrimination system for polynomials, new exact traveling wave solutions for generalized Ginzburg-Landau equation are obtained. The method has general meaning for many similar problems.
Exact solution of the Klein Gordon equation in the presence of a minimal length
Jana, T K
2009-01-01
We obtain exact solutions of the (1+1) dimensional Klein Gordon equation with linear vector and scalar potentials in the presence of a minimal length. Algebraic approach to the problem has also been studied.
Exact Solutions for a Coupled Korteweg-de Vries System
Zuo, Da-Wei; Jia, Hui-Xian
2016-11-01
Korteweg-de Vries (KdV)-type equation can be used to characterise the dynamic behaviours of the shallow water waves and interfacial waves in the two-layer fluid with gradually varying depth. In this article, by virtue of the bilinear forms, rational solutions and three kind shapes (soliton-like, kink and bell, anti-bell, and bell shapes) for the Nth-order soliton-like solutions of a coupled KdV system are derived. Propagation and interaction of the solitons are analyzed: (1) Potential u shows three kind of shapes (soliton-like, kink, and anti-bell shapes); Potential v exhibits two type of shapes (soliton-like and bell shapes); (2) Interaction of the potentials u and v both display the fusion phenomena.
Exact Scalar-Tensor Cosmological Solutions via Noether Symmetry
Belinchón, J A; Mak, M K
2016-01-01
In this paper, we investigate the Noether symmetries of a generalized scalar-tensor, Brans-Dicke type cosmological model, in which we consider explicit scalar field dependent couplings to the Ricci scalar, and to the scalar field kinetic energy, respectively. We also include the scalar field self-interaction potential into the gravitational action. From the condition of the vanishing of the Lie derivative of the gravitational cosmological Lagrangian with respect to a given vector field we obtain three cosmological solutions describing the time evolution of a spatially flat Friedman-Robertson-Walker Universe filled with a scalar field. The cosmological properties of the solutions are investigated in detail, and it is shown that they can describe a large variety of cosmological evolutions, including models that experience a smooth transition from a decelerating to an accelerating phase.
Exact Anisotropic Solutions of the Generalized TOV Equation
Riazi, Nematollah; Sajadi, S Naseh; Assyaee, S Shahrokh
2015-01-01
We explore gravitating relativistic spheres composed of an anisotropic, barotropic uid. We assume a bi-polytropic equation of state which has a linear and a power-law terms. The generalized Tolman-Oppenheimer-Volkoff (TOV) equation which describes the hydrostatic equilibrium is obtained. The full system of equations are solved for solutions which are regular at the origin and asymptotically flat. Conditions for the appearance of horizon and a basic treatment of stability are also discussed.
NEW EXACT SOLUTIONS TO KdV EQUATIONS WITH VARIABLE COEFFICIENTS OR FORCING
Institute of Scientific and Technical Information of China (English)
付遵涛; 刘式达; 刘式适; 赵强
2004-01-01
Jacobi elliptic function expansion method is extended to construct the exact solutions to another kind of KdV equations, which have variable coefficients or forcing terms. And new periodic solutions obtained by this method can be reduced to the solitontyped solutions under the limited condition.
An exact nonlinear hybrid-coordinate formulation for flexible multibody systems
Institute of Scientific and Technical Information of China (English)
Jinyang Liu; Jiazhen Hong; Lin Cui
2007-01-01
The previous low-order approximate nonlinear formulations succeeded in capturing the stiffening terms,but failed in simulation of mechanical systems with large deformation due to the neglect of the high-order deforma-tion terms. In this paper, a new hybrid-coordinate formu-lation is proposed, which is suitable for flexible multibody systems with large deformation. On the basis of exact strain-displacement relation, equations of motion for flexible multi-body system are derived by using virtual work principle. A matrix separation method is put forward to improve the efficiency of the calculation. Agreement of the present results with those obtained by absolute nodal coordinate formula-tion (ANCF) verifies the correctness of the proposed formu-lation. Furthermore, the present results are compared with those obtained by use of the linear model and the low-order approximate nonlinear model to show the suitability of the proposed models.
PAINLEV PROPERTY OF BURGERS-KDV EQUATION AND ITS EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we introduce the Painlev property of the Burgers-KdV equation. Two types of exact solutions to the equation are obtained by the standard truncated expansion metIn this paper, we introduce the Painlev property of the Burgers-KdV equation. Two types of exact solutions to the equation are obtained by the standard truncated expansion method and the extended standard truncated expansion method, respectively.hod and the extended standard truncated expansion method, respectively.
A new exact solution of the relativistic Boltzmann equation and its hydrodynamic limit
Denicol, Gabriel S; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael
2014-01-01
We present an exact solution of the relativistic Boltzmann equation for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse flow ("Gubser flow"). The resulting exact non-equilibrium dynamics is compared to 1st- and 2nd-order relativistic hydrodynamic approximations for various shear viscosity to entropy density ratios. This novel solution can be used to test the validity and accuracy of different hydrodynamic approximations in conditions similar to those generated in relativistic heavy-ion collisions.
Exact Solution of Unsteady Flow of Viscoelastic Fluid in a Pipe with Fractional Maxwell Model
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The unsteady flow of viscoelastic fluid in a cylindrical pipe was investigated using the fractional Maxwell model. Two special cases of unsteady pipe flow were expressed. The first is start-up flow, and the second is oscillating flow. The exact solution of start-up flow under a constant pressure gradient was obtained by using the theories of Laplace transform and Fourier-Bessel series for fractional derivatives. The exact solution of oscillating flow was obtained by utilizing the separation of variables.
Exact Solutions of Discrete Complex Cubic Ginzburg-Landau Equation and Their Linear Stability
Institute of Scientific and Technical Information of China (English)
张金良; 刘治国
2011-01-01
The discrete complex cubic Ginzburg-Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In this paper, the exact solutions of the discrete complex cubic Ginzburg-Landau equation are derived using homogeneous balance principle and the GI/G-expansion method, and the linear stability of exact solutions is discussed.
Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system
Cherniha, Roman
2010-01-01
Q-conditional symmetries of the classical Lotka-Volterra system in the case of one space variable are completely described and a set of such symmetries in explicit form is constructed. The relevant non-Lie ans\\"atze to reduce the classical Lotka-Volterra systems with correctly-specified coefficients to ODE systems and examples of new exact solutions are found. A possible biological interpretation of some exact solutions is presented.