若干非线性演化方程的精确解
|
摘要
随着科学技术的发展和线性理论的日趋完善,非线性科学已经在各个研究领域的作用越来越重要,逐步成为科学研究的焦点。物理、化学、生物、工程技术。甚至经济研究等都存在着大量的非线性问题,这些问题可以用非线性常微分方程或非线性偏微分方程来描述。至此,如何求解这些非线性方程无疑成为了非线性研究的关键和难点所在。
非线性方程不同于线性方程,虽然人们已经建立了不少行之有效的求解方法,如直接法,广义条件对称法,分离变量法,经典和非经典李群法,齐次平衡法等,但是,没有一种方法是能够普遍适用于所有的非线性方程的。所以,继续寻找一些有效可行的方法求解非线性方程仍然是一项十分重要的工作。
本文运用了一种指数函数展开法,进一步研究和分析了若干非线性演化方程,并得到了它们的更多的新的精确解。第一章中我们简介了几种求解非线性偏微分方程精确解的方法和非线性偏微分方程求解研究方法的发展状况。第二章中我们运用了指数函数展开法,求解了若干非线性演化方程(组),给出了它们更多的新的精确解。第三章我们对本文的工作做了总结和展望以及后续要研究的工作。
With the development of technology and the linear theory being perfect increasingly, the nonlinear science is becoming more and more important in almost all the scientific fields and close to the focus for the scientific research. There are a large of nonlinear problems that can be shown and described by the nonlinear ordinary differential equations and partial differential equations in physics. chemistry, biology, and also economics research.So,how to solve the nonlinear equations have becomed the crucial and difficult work for the researchers.
Nonlinear equations arc different from linear equations, although people have created many effective methods to deal with nonlinear equations like direct method, generalized condition symmetry, variable separation, classical and non-classical Lie group approaches, homogeneous balance method and so on, no one method can be suitable for all kinds of nonlinear equations. Then, it remains a very important work to look for new effective methods to deal with the nonlinear equations.
In this article,we utilize an exp(—φ(ξ))—expansion approach to study several nonlinear evolutional equations and get some new exact solutions of them. In chapter one, we introduce several methods to study the exact solution of nonlinear partial differential equations. In chapter two, we utilize an exp(—φ(ξ))—expansion approach to study several nonlinear evolutional equations and get some new exact solutions of them. In chapter three, we show the summary and expectation of this article.
非线性方程不同于线性方程,虽然人们已经建立了不少行之有效的求解方法,如直接法,广义条件对称法,分离变量法,经典和非经典李群法,齐次平衡法等,但是,没有一种方法是能够普遍适用于所有的非线性方程的。所以,继续寻找一些有效可行的方法求解非线性方程仍然是一项十分重要的工作。
本文运用了一种指数函数展开法,进一步研究和分析了若干非线性演化方程,并得到了它们的更多的新的精确解。第一章中我们简介了几种求解非线性偏微分方程精确解的方法和非线性偏微分方程求解研究方法的发展状况。第二章中我们运用了指数函数展开法,求解了若干非线性演化方程(组),给出了它们更多的新的精确解。第三章我们对本文的工作做了总结和展望以及后续要研究的工作。
With the development of technology and the linear theory being perfect increasingly, the nonlinear science is becoming more and more important in almost all the scientific fields and close to the focus for the scientific research. There are a large of nonlinear problems that can be shown and described by the nonlinear ordinary differential equations and partial differential equations in physics. chemistry, biology, and also economics research.So,how to solve the nonlinear equations have becomed the crucial and difficult work for the researchers.
Nonlinear equations arc different from linear equations, although people have created many effective methods to deal with nonlinear equations like direct method, generalized condition symmetry, variable separation, classical and non-classical Lie group approaches, homogeneous balance method and so on, no one method can be suitable for all kinds of nonlinear equations. Then, it remains a very important work to look for new effective methods to deal with the nonlinear equations.
In this article,we utilize an exp(—φ(ξ))—expansion approach to study several nonlinear evolutional equations and get some new exact solutions of them. In chapter one, we introduce several methods to study the exact solution of nonlinear partial differential equations. In chapter two, we utilize an exp(—φ(ξ))—expansion approach to study several nonlinear evolutional equations and get some new exact solutions of them. In chapter three, we show the summary and expectation of this article.
引文
[1]Ovler P.J..Application of Lie Groups to Differential Equation 2nd.,Graduate Texts Math.107 Springer,New York,1993
[2]Bluman G.W.,Kumei S..Symmetries and Differential Equation,Appl.Math.Sci.81 Springer,Berlin,1989
[3]Zhdanov R.Z..Conditional Lie-B(a|¨)cklund symmetry and reduction of evolution equation,J.Phys.A.,1991,28:3841-3850
[4]Zhdanov R.Z.,Lahno V.I..Conditional symmetry of a porous medium equation,Physics.D.,1998,122:178-186
[5]Qu C.Z.,He W.L.,Dou J.H..Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source,Physics D.,2000,144:97-123
[6]Qu C.Z.,He W.L.,Dou J.H..Separation of variables and exact solutions of generalized nonlinear Klein-Gordon equations,Prog.Theor.Phys.,2001,105:379-398
[7]Estevez P.G.,Qu C.Z.,Zhang S.L..Separation of variables of a generalized porous medium equation with nonlinear source.J.Math.Anal.Appl.,2002,275:44-59
[8]Estevez P.G.,Qu C.Z..Separation of variables in nonlinear wave equation with a variable wave speed,Theor.Math.Phys.,2002,133:1490-1497
[9]Lou S.Y.,Chen L.L..Formal variable separation approach for nonintegrable models,J.Math.Phys.,1999,40:6491-6500
[10]Dolye P.W.,Vassolion P.J..Separation of variables for the 1-dimensional nonlinear diffusion Equation,Int.J.Nonlinear Mech.,1998,33:315-326
[11]Dolye P.W..Separation of variable for scalar evolution equations in one space dimension,J.Phys.A:Math.Gen.,1996,29:7581-7595
[12]Kudryashow N.A..Exact solutions of the generalized kuramoto Sivashinsky equations[J].Physics Letters A,1990,147:287-292
[13]Liu S.K..Fu Z.T.,Liu S.D.,et al.A simple fast method in finding particular solutions of some nonlinear PDE[J].Applied Mathematics and Mechanics,2001,22(3):281-286
[14]Fu Z.T.,Liu S.D.,Liu S.K.,et al.New solutions to generalized mKdV equation [J].Connnunication in Theoretical Physics,2004,41:25-28
[15]Ablowitz M.J.,Clarkson P.A..Solitons,Nonlinear Evolution Equations and Inverse Scattering,Cambridge University Press,1991
[16]Gardner C.S.,Greene J.M.,Kruskal M.D.,et al.Method for solving the KdV equation,Phys.Rev.Lett.,1967,19:1095-1097
[17]谷超豪,胡和生,周子翔.孤子理论中的达布变换及其几何应用,上海科学技术出版社,2001
[18]Wadatiet ai R.I..Simpled erivation of B(a|¨)cklund transformation from Riccati form of inverse method,Prog.Theor.Phys.,1975,53:418-431
[19]Rogers C.,Schief W.K..B(a|¨)cklund and Darboux transformations geometry and modern applications in solition theory Cambridge University Press:Cambridge Texts in Applied Mathematics,2002
[20]Weiss J.,Tabor M.,Carmevale G..The painleve property for partial differential equations,J.Math.Phys.,1983,24:522-526
[21]Newell A.C.,Tabor M.,Zeng Y.B..A unified approach to painleve expansions,Phys.D.,1987,29:1-68
[22] Cariello F.,Tabor M.. painleve expansions for nonintegrable evolution equations, Phys. D., 1989,39:77-94
[23] Liu S.K..et al. Expansion method about the Jacobi elliptic function and its applications to nonlinear wave equations. Acta Physica Smica,200L50(ll):2068-2073(in Chinese)
[24] Fu Z.T.,et al. Solutions to Generalized mKdV equation. Communication in Theoretical Physics.2003,40(6):641-644
[25] Zhang G.X.,ct al.Exact solitary wave solutions of nonlinear wave equations. Science in China (Scries A)!2000,30(12):1103-1108 (in Chinese)
[26] Qu C.Z.,Zhang S.L.,Liu R.C..2000, Physica D., 144(97)
[27] Wang Mingliang. The solitary wave solutions for variant Boussinesq equations[J]. Physics Letter A,1995,199:169-172
[28] Wang Mingliang. Exact solutions for the RLW-Burgers equation [J]. Mathematica Applicata, 1995,8(l):51-55
[29] Wang Mingliang. Exact solutions for a compoud KdV-Burgers equation [J].Physics Letters A, 1996,213:279-287
[30] Liu Shikuo,Fu Zuntao,Liu Shida ,Zhao Qiang. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Phys.Lett.A, 2001,289:69-74
[31] Fu Zuntao,Liu Shikuo.Liu Shida,Zhao Qiang. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations [J]. Phys.Lett.A, 2001,290:72-76
[32] Wang Mingliang,Zhou Yubin. The periodic wave solutions for the Klein -Gordon - Shrodinger equation [J]. Phys.Lett.A, 2003,318:84-92
[33]Zhou Yubin,Wang Mingliang,Wang Yueming.Periodic wave solutions to a coupled KdV equations with variable coefficients[J].Phys.Lett.A,2003,308:31-36
[34]Zhou Yubin,Wang Mingliang,Miao Tiande.The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations [J].Phys.Lett.A,2004,323:77-88
[35]He J.H.,Wu X.H..Exp-funtion method for nonlinear wave equations[J].Chaos,Solutions and Fractals,2006,30(3):700-708
[35]Zhao Meimei,Li Chao.The exp(-φ(ξ))-expansion Method applied to Nonlinear Evolution Equations http://www.paper.edu.cn/paper.php?serialnumber =200805-762
[37]Yue D.,Won S..Delay dependent robust stability of stochastic systems with time delay and nonlinear uncertainties[J].Electronics Letters,2001,37(15):992-993
[38]Shen Shoufeng.Exact Solutions of Some(1+1)-Dimensional Nonlinear Evolution Equations.Commun.Theor,Phys.,(Beijing,Chian)2006(45):236-238
[39]Hirota R,Satuma J.Soliton solutions of a coupled KdV equation.Phys.Lett.A,1981,85:407-416
[40]Zhang Xuedong,Chen Yong,Zhang Hongqing.Generalized Extended Tanhfuntion Method and Its Application to(1+1)-dimensional Dispersive Long Wave Equation[J].Physics Letter A,2003,311:145-157
[41]楼森岳,唐晓艳.非线性数学物理方法(第一版),科学出版社,2006
[42]谷超豪,李大潜,陈恕行,郑宋穆,潭永基.数学物理方程(第二版),高等教育出版社,2005
[2]Bluman G.W.,Kumei S..Symmetries and Differential Equation,Appl.Math.Sci.81 Springer,Berlin,1989
[3]Zhdanov R.Z..Conditional Lie-B(a|¨)cklund symmetry and reduction of evolution equation,J.Phys.A.,1991,28:3841-3850
[4]Zhdanov R.Z.,Lahno V.I..Conditional symmetry of a porous medium equation,Physics.D.,1998,122:178-186
[5]Qu C.Z.,He W.L.,Dou J.H..Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source,Physics D.,2000,144:97-123
[6]Qu C.Z.,He W.L.,Dou J.H..Separation of variables and exact solutions of generalized nonlinear Klein-Gordon equations,Prog.Theor.Phys.,2001,105:379-398
[7]Estevez P.G.,Qu C.Z.,Zhang S.L..Separation of variables of a generalized porous medium equation with nonlinear source.J.Math.Anal.Appl.,2002,275:44-59
[8]Estevez P.G.,Qu C.Z..Separation of variables in nonlinear wave equation with a variable wave speed,Theor.Math.Phys.,2002,133:1490-1497
[9]Lou S.Y.,Chen L.L..Formal variable separation approach for nonintegrable models,J.Math.Phys.,1999,40:6491-6500
[10]Dolye P.W.,Vassolion P.J..Separation of variables for the 1-dimensional nonlinear diffusion Equation,Int.J.Nonlinear Mech.,1998,33:315-326
[11]Dolye P.W..Separation of variable for scalar evolution equations in one space dimension,J.Phys.A:Math.Gen.,1996,29:7581-7595
[12]Kudryashow N.A..Exact solutions of the generalized kuramoto Sivashinsky equations[J].Physics Letters A,1990,147:287-292
[13]Liu S.K..Fu Z.T.,Liu S.D.,et al.A simple fast method in finding particular solutions of some nonlinear PDE[J].Applied Mathematics and Mechanics,2001,22(3):281-286
[14]Fu Z.T.,Liu S.D.,Liu S.K.,et al.New solutions to generalized mKdV equation [J].Connnunication in Theoretical Physics,2004,41:25-28
[15]Ablowitz M.J.,Clarkson P.A..Solitons,Nonlinear Evolution Equations and Inverse Scattering,Cambridge University Press,1991
[16]Gardner C.S.,Greene J.M.,Kruskal M.D.,et al.Method for solving the KdV equation,Phys.Rev.Lett.,1967,19:1095-1097
[17]谷超豪,胡和生,周子翔.孤子理论中的达布变换及其几何应用,上海科学技术出版社,2001
[18]Wadatiet ai R.I..Simpled erivation of B(a|¨)cklund transformation from Riccati form of inverse method,Prog.Theor.Phys.,1975,53:418-431
[19]Rogers C.,Schief W.K..B(a|¨)cklund and Darboux transformations geometry and modern applications in solition theory Cambridge University Press:Cambridge Texts in Applied Mathematics,2002
[20]Weiss J.,Tabor M.,Carmevale G..The painleve property for partial differential equations,J.Math.Phys.,1983,24:522-526
[21]Newell A.C.,Tabor M.,Zeng Y.B..A unified approach to painleve expansions,Phys.D.,1987,29:1-68
[22] Cariello F.,Tabor M.. painleve expansions for nonintegrable evolution equations, Phys. D., 1989,39:77-94
[23] Liu S.K..et al. Expansion method about the Jacobi elliptic function and its applications to nonlinear wave equations. Acta Physica Smica,200L50(ll):2068-2073(in Chinese)
[24] Fu Z.T.,et al. Solutions to Generalized mKdV equation. Communication in Theoretical Physics.2003,40(6):641-644
[25] Zhang G.X.,ct al.Exact solitary wave solutions of nonlinear wave equations. Science in China (Scries A)!2000,30(12):1103-1108 (in Chinese)
[26] Qu C.Z.,Zhang S.L.,Liu R.C..2000, Physica D., 144(97)
[27] Wang Mingliang. The solitary wave solutions for variant Boussinesq equations[J]. Physics Letter A,1995,199:169-172
[28] Wang Mingliang. Exact solutions for the RLW-Burgers equation [J]. Mathematica Applicata, 1995,8(l):51-55
[29] Wang Mingliang. Exact solutions for a compoud KdV-Burgers equation [J].Physics Letters A, 1996,213:279-287
[30] Liu Shikuo,Fu Zuntao,Liu Shida ,Zhao Qiang. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Phys.Lett.A, 2001,289:69-74
[31] Fu Zuntao,Liu Shikuo.Liu Shida,Zhao Qiang. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations [J]. Phys.Lett.A, 2001,290:72-76
[32] Wang Mingliang,Zhou Yubin. The periodic wave solutions for the Klein -Gordon - Shrodinger equation [J]. Phys.Lett.A, 2003,318:84-92
[33]Zhou Yubin,Wang Mingliang,Wang Yueming.Periodic wave solutions to a coupled KdV equations with variable coefficients[J].Phys.Lett.A,2003,308:31-36
[34]Zhou Yubin,Wang Mingliang,Miao Tiande.The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations [J].Phys.Lett.A,2004,323:77-88
[35]He J.H.,Wu X.H..Exp-funtion method for nonlinear wave equations[J].Chaos,Solutions and Fractals,2006,30(3):700-708
[35]Zhao Meimei,Li Chao.The exp(-φ(ξ))-expansion Method applied to Nonlinear Evolution Equations http://www.paper.edu.cn/paper.php?serialnumber =200805-762
[37]Yue D.,Won S..Delay dependent robust stability of stochastic systems with time delay and nonlinear uncertainties[J].Electronics Letters,2001,37(15):992-993
[38]Shen Shoufeng.Exact Solutions of Some(1+1)-Dimensional Nonlinear Evolution Equations.Commun.Theor,Phys.,(Beijing,Chian)2006(45):236-238
[39]Hirota R,Satuma J.Soliton solutions of a coupled KdV equation.Phys.Lett.A,1981,85:407-416
[40]Zhang Xuedong,Chen Yong,Zhang Hongqing.Generalized Extended Tanhfuntion Method and Its Application to(1+1)-dimensional Dispersive Long Wave Equation[J].Physics Letter A,2003,311:145-157
[41]楼森岳,唐晓艳.非线性数学物理方法(第一版),科学出版社,2006
[42]谷超豪,李大潜,陈恕行,郑宋穆,潭永基.数学物理方程(第二版),高等教育出版社,2005