非线性偏微分方程的几类求解方法
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摘要
在齐次平衡原则的思想下,充分利用F-展开法、双曲正切函数法和Riccati方程在非线性偏微分方程(NLPDES)求解中的优良特性,提出一种广义扩展的F-展开法和改进的双曲正切函数法。此方法在借助于计算机符号系统Mathematica下,操作方便,可以得到NLPDES的一系列精确解如周期波解、类孤子解、三角函数解、有理函数解、复数形式解,扭曲状解等。并利用广义扩展的F-展开法求解了非线性色散耗散mKdV方程,利用改进的双曲正切函数法求解了(3+1)维Burgers方程,得到了它们类型丰富的精确解,其中部分是新解。文中对部分解进行了数值模拟以便直观分析。
首先,在齐次平衡思想的基础上利用改进的辅助方程方法求解Klein-Gordon方程,得到了更多丰富类型的Klein-Gordon方程的行波解如类孤子解,三角函数周期解,有理数解,指数解。
其次,利用广义扩展的F-展开法研究了非线性色散耗散mKdV方程,得到了非线性色散耗散mKdV方程的周期波解、类孤子解、三角函数解、有理函数解、复数形式解等。这些解对于解释一些物理现象具有一定的意义。
最后,利用改进的双曲正切函数法研究了(3+1)维Burgers方程,得到了他们类型丰富的精确解:光滑的钟形孤立波解,kink解,类孤子解,复数形式解,有理数解等,并得到了部分新解。这对于对非线性偏微分方程的进一步研究具有积极的意义。
Under the Homogeneous balance idea, the generalized modified F-expansion method and the extended modified tanh-function method are proposed by taking full advantages of F-expansion method, tanh-function method and Riccati equation in seeking exact solutions of NLPDEs. The method can be conveniently operated with the aid of computer symbolic systems Mathematica, and rich families of exact solutions of NLPDEs have been obtained, including periodic wave solutions, solitary wave solutions, triangle function solutions, rational function solutions, plural number formal solutions, bell-shaped solitary solutions, kink-shape solutions and so on. By using the generalized modified F-expansion method and the extended modified tanh-function method, we have solved the nonlinear dispersive dissipative mKdV equation and the (3+1)-dimensional Burgers equations respectively. Massive exact solutions of them have been obtained, and some of the solutions are new. We also provided some figures of partial solutions for direct-viewing analysis.
Firstly, we researched the Klein-Gordon equation by using the modified auxiliary method under Homogeneous balance idea. As a result, many new and more general exact traveling wave solutions are obtained, such as soliton-like solutions, trigonometric function solutions, exponential solutions and rational solutions, etc.
Next, we researched the exact solutions of the nonlinear dispersive dissipative mKdV equation by using a generalized modified F-expansion method. Rich families of exact solutions of them have been obtained, including periodic wave solutions, soliton-like solutions, triangle function solutions, rational function solutions, plural number formal solutions and so on. And some of them are new. We consider these solutions will make sense for explaining some physical phenomenon.
Finally, we used the extended modified tanh-function method to solve the (3+1)-dimensional Burgers equations and obtained more rich exact solutions of them, like bell-shaped solitary solutions, kink solutions, soliton-like solutions, plural number formal solutions, rational solutions and so on. This work will be usefull for further research to the equation.
首先,在齐次平衡思想的基础上利用改进的辅助方程方法求解Klein-Gordon方程,得到了更多丰富类型的Klein-Gordon方程的行波解如类孤子解,三角函数周期解,有理数解,指数解。
其次,利用广义扩展的F-展开法研究了非线性色散耗散mKdV方程,得到了非线性色散耗散mKdV方程的周期波解、类孤子解、三角函数解、有理函数解、复数形式解等。这些解对于解释一些物理现象具有一定的意义。
最后,利用改进的双曲正切函数法研究了(3+1)维Burgers方程,得到了他们类型丰富的精确解:光滑的钟形孤立波解,kink解,类孤子解,复数形式解,有理数解等,并得到了部分新解。这对于对非线性偏微分方程的进一步研究具有积极的意义。
Under the Homogeneous balance idea, the generalized modified F-expansion method and the extended modified tanh-function method are proposed by taking full advantages of F-expansion method, tanh-function method and Riccati equation in seeking exact solutions of NLPDEs. The method can be conveniently operated with the aid of computer symbolic systems Mathematica, and rich families of exact solutions of NLPDEs have been obtained, including periodic wave solutions, solitary wave solutions, triangle function solutions, rational function solutions, plural number formal solutions, bell-shaped solitary solutions, kink-shape solutions and so on. By using the generalized modified F-expansion method and the extended modified tanh-function method, we have solved the nonlinear dispersive dissipative mKdV equation and the (3+1)-dimensional Burgers equations respectively. Massive exact solutions of them have been obtained, and some of the solutions are new. We also provided some figures of partial solutions for direct-viewing analysis.
Firstly, we researched the Klein-Gordon equation by using the modified auxiliary method under Homogeneous balance idea. As a result, many new and more general exact traveling wave solutions are obtained, such as soliton-like solutions, trigonometric function solutions, exponential solutions and rational solutions, etc.
Next, we researched the exact solutions of the nonlinear dispersive dissipative mKdV equation by using a generalized modified F-expansion method. Rich families of exact solutions of them have been obtained, including periodic wave solutions, soliton-like solutions, triangle function solutions, rational function solutions, plural number formal solutions and so on. And some of them are new. We consider these solutions will make sense for explaining some physical phenomenon.
Finally, we used the extended modified tanh-function method to solve the (3+1)-dimensional Burgers equations and obtained more rich exact solutions of them, like bell-shaped solitary solutions, kink solutions, soliton-like solutions, plural number formal solutions, rational solutions and so on. This work will be usefull for further research to the equation.
引文
[1]陈陆君,梁昌洪.孤立子理论及其应用,西安电子科技大学出版社,1997
[2]倪皖荪,魏荣爵.水槽中的孤波,上海科技教育出版社,1997
[3]李翊神.孤子与可积系统,上海科技教育出版社,1999
[4]谷超豪等.孤立子理论与应用,浙江科学技术出版社,1990
[5]郭柏灵,苏凤秋.孤立子,辽宁教育出版社,1998.
[6]黄景宁,徐济仲,熊吟涛.孤子概念、原理和应用,高等教育出版社,2004.
[7]刘式达,刘式适.孤波和湍流,上海科技教育出版社,1994.
[8]Ablowitz M J,Clarkson P A.孤立子、非线性发展方程和逆散射,世界图书出版公司北京公司,2000.
[9]刘正荣.关于孤立子的研究.云南大学学报,2003;25:207-211.
[10]Abolowitz M J,Clarkson P A.Solitons,Nonlinear evolution equations and inverse scatting.London:Cambridge University Press,1991.
[11]田畴.李群及其在微分方程中的应用.科学出版社,2001.
[12]Gardner C S,Greene J M,Kruskal M D,Miura R M.Method for solving the Korteweg-de Vires equation.Phys.Rev.Lett,1967;19:1095-1097.
[13]Lax P D.Integrals of nonlinear equations of evolution and solitary waves.Commun.Pure.Appl.Math,1968;21:467-490.
[14]Zakharov V E,Shabat A B.Exact theory of two-dimensional self-focusing and one-dimensional of waves in nonlinear media.Sov.Phys.Jetp,1972;34:62-69.
[15]Ablowitz M J,Kaup D J,Newell A C,Segur H.Methods for solving the sine-Gordon equation.,Phy.Rev.Lett.1973,30:1262-1264
[16]Ablowitz M J,Kaup D J,Newell A C,Segur H.Nonlinear evolution equations of physical significance.Phy.Rev.Lett.1974,31:125-127
[17]Ablowitz M J,Kaup D J,Newell A C,Segur H.The ineverse scattering transform-Fourier analysis for nonlinear problems.Stud.in.Appl.Math,1974;53:249-315
[18]Hirota R.Exact solution of the Korteweg-de Vries equation for multipile collisions of solutions.Phys.Rev.Lett.1971,27:1192-1194
[19]Hirota R.Direct method of finding exact solution of nonlinear evolution equations,in "B(a|¨)cklund transformation".Edited by Miura R M.Lecture Notes in Mathematics,Springer,Berlin.1976:515
[20]Fan E G.Extended tanh-function method and its applications to nonlinear equations.Phys Lett A,2000;277:212-218.
[21]Abdusalam H A.On an improved complex tanh-function method.Nonlinear Sciences and Numerical Simulation,2005;6(2):99-106.
[22]Yah CT.A simple transformation for nonlinear waves.Phys Lett A,1996;224:77-84.
[23]付遵涛,刘式适,刘式达.非线性波方程求解的新方法.物理学报,2004,53(2):343-348.
[24]Chen Y,Li B.General projective Riccati equation method and exact solutions for generalized KdV-type and KdV-Bergers-type equations with nonlinear terms of any order.Chaos,Solitons and Fractals,2004;19(4):977-984.
[25]Chen Y,Wang Q.Multiple Riccati equations rational expansion method and complexion solutions of the Whitham-Broer-Kaup equation.Phys Lett A,2005;347:215-227.
[26]Zhou YB,Wang ML,Wang YM.Periodic wave solutions to a coupled KdV equation with variable coefficients.Phys Lett A,2003;308:31-36.
[27]Liu JB,Yang KQ.The extended F-expansion method and exact solutions of nonlinear PDEs [J].Chaos,Solitons and Fractals,2004;22(1):111-121.
[28]Wang DS,Zhang HQ.Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation.Chaos,Solitons and Fractals,2005;25:601-610.
[29]Yomba E.The extended F-expansion method and its application,for solving the nonlinear wave,CKGZ,GDS,DS and GZ equations.Phys Lett A,2005;340:149-160.
[30]Shen J W,Miao B J and Guo L L.Bifurcation analysis of travelling wave solutions in the nonlinear Klein-Gordon model with anharmonic coupling.Appl.Math.Comp.2007;188:1975-1983.
[31]Tang X Y,Shukla P K.Periodic travelling and non-travelling wave solutions of the nonlinear Klein-Gordon equation with imaginary mass.Phys.Lett.A.2008;372:258-262.
[32]Sirendaoreji.Exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations.Phys.Lett.A.2007;363:440-447.
[33]Yang C D.On the existence of complex spacetime in relativistic quantum mechanics.Chaos,Soliton and Fractals,2008;38:316-331.
[34]Dmitriev S V.Discrete systems free of the Peierls-Nabarro potential.Non-Crystalline Solids,2008;354:4121-4125.
[35]Wang S S,Zhang L M.A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schr(o|¨)dinger equations.Appl.Math.Comp,2008;203:799-812.
[36]Konoplya R A.Superradiant instability for black holes immersed in a magnetic field.Phys Lett B,2008;666:283-287.
[37]Anjan B,Chenwi Z,Essaid Z.Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation.Appl.Math.Comp,2008;203:153-156.
[38]Sirendaoreji.A new anxiliary equation and exact traveling wave solutiomns of nonlinear equations.Phys Lett A,2006;356:124-130.
[39]Kappeler T,Perry P,Shubin M.Solutions of mKdv in classes of functions unbounded at infinity.Journal of Geometric Analysis,2008,18(2):443-477.
[40]张金良,王明亮,王跃明.推广的F-展开法及变系数KdV和mKdv的精确解.物理学报,2006;26A(3):353-360.
[41]Tian Lixin,Yin Jiuli.Multi-compacton and double symmetric peakon for generalized Ostrovsky equation.Chaos Solitons and Fractals,2008,35:991-995.
[42]MaS H,Wu X H,Fang J P.New exact solutions and special soliton structures for the(3+1)-dimensional Burgers system.Chin.Phys.Soci.57(2008),pp.11-17.
[43]Zhu H P,Pan Z H,Zheng C L.Embedded-Soliton and Complex Wave Excitations of (3+1)-Dimensional Burgers System.Commun.Theor.Phys.2008;49:1425-1431.
[44]Dai C Q,Yan C J,Zhang J F.Variable Separation Solutions in(1+1)-dimensional and (3+1)-dimensional Systems via Entangled Mapping Approach.Commun.Theor.Phys.2006;46:389-392.
[45]Gandarias M L.Type-Ⅱ hidden symmetries through weak symmetries for nonlinear partial differential equations.Math.Anal.Appl.2008;348:752-759.
[46]Wang B D,Song L N,Zhang H Q.A new extended eIliptic equation rational expansion method and its.application to the(2+1)-dimensional Burgers equations.Chaos,Soliton and Fractals,2007;33:1546-1551.
[47]Zhi H Y,Zhang H Q.New rational solitary wave solutions of(2+1)-dimensional Burgers equation.Nonlinear Analysis,2007;66:2264-2273.
[48]Wang Q,Song L N,Zhang H Q.A new coupled sub-equations expansion method and novel complexiton solutions of(2+1)-dimensional Burgers equation.Appl.Math.Comp,2007;186:632-637.
[49]Wang Q,Song L N,Zhang H Q.A new coupled sub-equations expansion method and novel complexiton solutions of(2+1)-dimensional Burgers equation.Appl.Math.Comp,2007;186:632-637.
[50]Sarrico C O R.New solutions for the one-dimensional nonconservative inviscid Burgers equation.Math.Anal.Appl.2006;317:496-509.
[2]倪皖荪,魏荣爵.水槽中的孤波,上海科技教育出版社,1997
[3]李翊神.孤子与可积系统,上海科技教育出版社,1999
[4]谷超豪等.孤立子理论与应用,浙江科学技术出版社,1990
[5]郭柏灵,苏凤秋.孤立子,辽宁教育出版社,1998.
[6]黄景宁,徐济仲,熊吟涛.孤子概念、原理和应用,高等教育出版社,2004.
[7]刘式达,刘式适.孤波和湍流,上海科技教育出版社,1994.
[8]Ablowitz M J,Clarkson P A.孤立子、非线性发展方程和逆散射,世界图书出版公司北京公司,2000.
[9]刘正荣.关于孤立子的研究.云南大学学报,2003;25:207-211.
[10]Abolowitz M J,Clarkson P A.Solitons,Nonlinear evolution equations and inverse scatting.London:Cambridge University Press,1991.
[11]田畴.李群及其在微分方程中的应用.科学出版社,2001.
[12]Gardner C S,Greene J M,Kruskal M D,Miura R M.Method for solving the Korteweg-de Vires equation.Phys.Rev.Lett,1967;19:1095-1097.
[13]Lax P D.Integrals of nonlinear equations of evolution and solitary waves.Commun.Pure.Appl.Math,1968;21:467-490.
[14]Zakharov V E,Shabat A B.Exact theory of two-dimensional self-focusing and one-dimensional of waves in nonlinear media.Sov.Phys.Jetp,1972;34:62-69.
[15]Ablowitz M J,Kaup D J,Newell A C,Segur H.Methods for solving the sine-Gordon equation.,Phy.Rev.Lett.1973,30:1262-1264
[16]Ablowitz M J,Kaup D J,Newell A C,Segur H.Nonlinear evolution equations of physical significance.Phy.Rev.Lett.1974,31:125-127
[17]Ablowitz M J,Kaup D J,Newell A C,Segur H.The ineverse scattering transform-Fourier analysis for nonlinear problems.Stud.in.Appl.Math,1974;53:249-315
[18]Hirota R.Exact solution of the Korteweg-de Vries equation for multipile collisions of solutions.Phys.Rev.Lett.1971,27:1192-1194
[19]Hirota R.Direct method of finding exact solution of nonlinear evolution equations,in "B(a|¨)cklund transformation".Edited by Miura R M.Lecture Notes in Mathematics,Springer,Berlin.1976:515
[20]Fan E G.Extended tanh-function method and its applications to nonlinear equations.Phys Lett A,2000;277:212-218.
[21]Abdusalam H A.On an improved complex tanh-function method.Nonlinear Sciences and Numerical Simulation,2005;6(2):99-106.
[22]Yah CT.A simple transformation for nonlinear waves.Phys Lett A,1996;224:77-84.
[23]付遵涛,刘式适,刘式达.非线性波方程求解的新方法.物理学报,2004,53(2):343-348.
[24]Chen Y,Li B.General projective Riccati equation method and exact solutions for generalized KdV-type and KdV-Bergers-type equations with nonlinear terms of any order.Chaos,Solitons and Fractals,2004;19(4):977-984.
[25]Chen Y,Wang Q.Multiple Riccati equations rational expansion method and complexion solutions of the Whitham-Broer-Kaup equation.Phys Lett A,2005;347:215-227.
[26]Zhou YB,Wang ML,Wang YM.Periodic wave solutions to a coupled KdV equation with variable coefficients.Phys Lett A,2003;308:31-36.
[27]Liu JB,Yang KQ.The extended F-expansion method and exact solutions of nonlinear PDEs [J].Chaos,Solitons and Fractals,2004;22(1):111-121.
[28]Wang DS,Zhang HQ.Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation.Chaos,Solitons and Fractals,2005;25:601-610.
[29]Yomba E.The extended F-expansion method and its application,for solving the nonlinear wave,CKGZ,GDS,DS and GZ equations.Phys Lett A,2005;340:149-160.
[30]Shen J W,Miao B J and Guo L L.Bifurcation analysis of travelling wave solutions in the nonlinear Klein-Gordon model with anharmonic coupling.Appl.Math.Comp.2007;188:1975-1983.
[31]Tang X Y,Shukla P K.Periodic travelling and non-travelling wave solutions of the nonlinear Klein-Gordon equation with imaginary mass.Phys.Lett.A.2008;372:258-262.
[32]Sirendaoreji.Exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations.Phys.Lett.A.2007;363:440-447.
[33]Yang C D.On the existence of complex spacetime in relativistic quantum mechanics.Chaos,Soliton and Fractals,2008;38:316-331.
[34]Dmitriev S V.Discrete systems free of the Peierls-Nabarro potential.Non-Crystalline Solids,2008;354:4121-4125.
[35]Wang S S,Zhang L M.A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schr(o|¨)dinger equations.Appl.Math.Comp,2008;203:799-812.
[36]Konoplya R A.Superradiant instability for black holes immersed in a magnetic field.Phys Lett B,2008;666:283-287.
[37]Anjan B,Chenwi Z,Essaid Z.Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation.Appl.Math.Comp,2008;203:153-156.
[38]Sirendaoreji.A new anxiliary equation and exact traveling wave solutiomns of nonlinear equations.Phys Lett A,2006;356:124-130.
[39]Kappeler T,Perry P,Shubin M.Solutions of mKdv in classes of functions unbounded at infinity.Journal of Geometric Analysis,2008,18(2):443-477.
[40]张金良,王明亮,王跃明.推广的F-展开法及变系数KdV和mKdv的精确解.物理学报,2006;26A(3):353-360.
[41]Tian Lixin,Yin Jiuli.Multi-compacton and double symmetric peakon for generalized Ostrovsky equation.Chaos Solitons and Fractals,2008,35:991-995.
[42]MaS H,Wu X H,Fang J P.New exact solutions and special soliton structures for the(3+1)-dimensional Burgers system.Chin.Phys.Soci.57(2008),pp.11-17.
[43]Zhu H P,Pan Z H,Zheng C L.Embedded-Soliton and Complex Wave Excitations of (3+1)-Dimensional Burgers System.Commun.Theor.Phys.2008;49:1425-1431.
[44]Dai C Q,Yan C J,Zhang J F.Variable Separation Solutions in(1+1)-dimensional and (3+1)-dimensional Systems via Entangled Mapping Approach.Commun.Theor.Phys.2006;46:389-392.
[45]Gandarias M L.Type-Ⅱ hidden symmetries through weak symmetries for nonlinear partial differential equations.Math.Anal.Appl.2008;348:752-759.
[46]Wang B D,Song L N,Zhang H Q.A new extended eIliptic equation rational expansion method and its.application to the(2+1)-dimensional Burgers equations.Chaos,Soliton and Fractals,2007;33:1546-1551.
[47]Zhi H Y,Zhang H Q.New rational solitary wave solutions of(2+1)-dimensional Burgers equation.Nonlinear Analysis,2007;66:2264-2273.
[48]Wang Q,Song L N,Zhang H Q.A new coupled sub-equations expansion method and novel complexiton solutions of(2+1)-dimensional Burgers equation.Appl.Math.Comp,2007;186:632-637.
[49]Wang Q,Song L N,Zhang H Q.A new coupled sub-equations expansion method and novel complexiton solutions of(2+1)-dimensional Burgers equation.Appl.Math.Comp,2007;186:632-637.
[50]Sarrico C O R.New solutions for the one-dimensional nonconservative inviscid Burgers equation.Math.Anal.Appl.2006;317:496-509.