用标准的和推广的tanh函数展开法求解非线性发展方程
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摘要
随着非线性科学的不断发展,人们对非线性现象的研究越来越深入,研究的领域也越来越广泛,比如：非线性发展方程解的求法以及解的存在性、唯一性、稳定性等.其中对精确解的研究始终是一个重要课题,虽然现在已经形成了许多求解非线性发展方程的方法,但这些方法都仅适用于某些类型方程,对所有的非线性发展方程,没有普遍适用的方法.
本文利用标准的和推广的tanh函数展开法对一些非线性发展方程进行求解,得到了大量的解,充分展示了tanh函数展开法的有效性和实用性.
本文的内容和结构安排如下：
第一章介绍非线性发展方程精确解的几种常见解法,并简单地回顾了精确解研究的发展概况.
第二章用标准的tanh函数展开法对一些非线性发展方程进行求解,包括(3+1)维的KP方程、sine-Gordon方程、Higher-order KdV-like方程和Klein-Gordon-type方程.
第三章利用推广的tanh函数展开法求解Boussinesq-type方程和breaking方程.
最后,对本文的工作进行总结与展望,并且提出了一些自己有待研究的问题.
With the development of nonlinear science, the study of nonlinear phenomena is becoming more and more clear, and the field of the study becomes very wide, such as the method to solve evolution equations and the method to discuss the existence, uniqueness, stability of exact solutions. The study of exact solutions is an important topic. There are a number of methods to solve evolution equations, but they are only applicable to certain types of equations. In fact, there is no method can be worked on all nonlinear evolution equations.
This paper, we use a standard tanh-function method and the extended tanh-function method to solve some nonlinear evolution equations, and obtain a large number of exact solutions. It shows the effectiveness of tanh-function method.
The structure of this paper is as follows:
Chapter 1, we introduce several common methods to solve nonlinear evolution equations, and briefly review the present condition for the study of exact solutions.
Chapter 2, we use the standard tanh-function method to solve several nonlinear evolution equations, which include (3+1)-dimensional KP, sine-Gordon, Higher-order KdV-like and Klein-Gordon-type equations.
Chapter 3, exact solutions of Boussinesq-type equation and breaking equation are obtained by using extended tanh-function method.
In the end, we summarize the work of this paper, and bring up a number of problems to be solved in the future.
本文利用标准的和推广的tanh函数展开法对一些非线性发展方程进行求解,得到了大量的解,充分展示了tanh函数展开法的有效性和实用性.
本文的内容和结构安排如下：
第一章介绍非线性发展方程精确解的几种常见解法,并简单地回顾了精确解研究的发展概况.
第二章用标准的tanh函数展开法对一些非线性发展方程进行求解,包括(3+1)维的KP方程、sine-Gordon方程、Higher-order KdV-like方程和Klein-Gordon-type方程.
第三章利用推广的tanh函数展开法求解Boussinesq-type方程和breaking方程.
最后,对本文的工作进行总结与展望,并且提出了一些自己有待研究的问题.
With the development of nonlinear science, the study of nonlinear phenomena is becoming more and more clear, and the field of the study becomes very wide, such as the method to solve evolution equations and the method to discuss the existence, uniqueness, stability of exact solutions. The study of exact solutions is an important topic. There are a number of methods to solve evolution equations, but they are only applicable to certain types of equations. In fact, there is no method can be worked on all nonlinear evolution equations.
This paper, we use a standard tanh-function method and the extended tanh-function method to solve some nonlinear evolution equations, and obtain a large number of exact solutions. It shows the effectiveness of tanh-function method.
The structure of this paper is as follows:
Chapter 1, we introduce several common methods to solve nonlinear evolution equations, and briefly review the present condition for the study of exact solutions.
Chapter 2, we use the standard tanh-function method to solve several nonlinear evolution equations, which include (3+1)-dimensional KP, sine-Gordon, Higher-order KdV-like and Klein-Gordon-type equations.
Chapter 3, exact solutions of Boussinesq-type equation and breaking equation are obtained by using extended tanh-function method.
In the end, we summarize the work of this paper, and bring up a number of problems to be solved in the future.
引文
[1]付遵涛,刘式适,刘式达.非线性波方程求解的新方法[J].物理学报,2004,53(2)：343-348
[2]Liu S. K., Fu Z. T., Liu S. D.. Jacobi Elliptic Function Expansion Method and Pe riodic Wave Solutions of Nonlinear Wave Equations[J]. Physics Letters A,2001, V 28(9):69-74
[3]Filiz T., Ahmet B., Murat K.. Travelling Wave Solution of Nonlinear Evolution Eq uation by Using the First Integral Method[J]. Commun Nonlinear Sci Number Sim ular,2009, V14:1810-1815
[4]Feng Z. S.. The First Intergral Method to Study the Burgers-Korteweg-deVriesequat ion[J]. J. Phys. A,2002, V35:343-349
[5]Feng Z. S.. Exact Solution to an Approximate sine-Gordon Equation in (n+l)dime nsional Space[J]. Phys. Lett. A,2002, V302:64-76
[6]Feng Z. S., Wang X. H.. The First Intergral Method to the Two-dimensional Burg ers-Korteweg-deVriesequation[J]. Phys. Lett. A,2003, V308:173-178
[7]Naranmandula M., Wang K. X.. New Explicit Exact Solutions to a Nonlinear Disp ersive-dissipative Equation[J]. Chinese Physics,2004, V13(2):139-143
[8]Inc M.. New Exact Solutions for the ZK-MEW Equation by Using Symbolic Com-putation[J]. Appl. Math. Comput,2007, V189(508)
[9]He J., Wu X.. Exp-function Method for Nonlinear Wave Equations Chao Solutions and Fractals[J]. Phys. Lett. A,2006, V30(3):700-708
[10]Zhao M., Li C.. The Expansion Method Applied to Nonlinear Evolution Equation. http://www.paper.edu.cn/paper.php?serial number=200805-762
[11]祁新雷.若干非线性演化方程的精确解[D].西安：西北大学,2009
[12]史良马.变系数非线性方程的求解[D].安徽：安徽大学,2006
[13]Ablowitz M., Clarkson P.. Solutions Nonlinear Evolutions and Inverse Scattering Transform[M]. Cambridg University Press,1990
[14]Vakhnenko V., parkes E., Morrison A.. A Backiund Transformations and the Inver se Scattering Transform Method for the Generalized Vakhenko Equation Chaos sol ution Fract[J]. J. Phys. A,2003, V17(4):683-685
[15]Gardner C., Greene J., Kruskal M., etal. Method for Solving the Kortewege-de V ries Equation[J]. Phys. Lett.,1967, V19:1095-1907
[16]Hirota R.. A New Form of Backlund Transformations and Its Relation to the Inv erse Scattering Problem[J]. Prog. Theor. Phys.,1974, V52(5):1498-1512
[17]Lamb G.. Backlund Transformations for Certain Nonlinear Evolution Equations[J]. Math. phys.,1974, V15:2157-2165
[18]Wu H.. On Backlund Transformations for Nonlinear Partial Differential Equations [J].J. Math. Anal. Appl.,1995, V192:151-179
[19]Matveev V.. Darbous Transformations and Soliton Solutions[R]. Berlin,1991
[20]曹生让.几个非线性演化方程的精确解[D].江苏：江苏大学,2007
[21]李德生.若干非线性演化方程精确求解法的研究[D].大连：大连理工,2004
[22]Weiss J., Tabor M., Carnevale G. et al. The Painleve Property for Partica Differe-ntial Equations[J]. Math. Phys.,1983, V24:522-526
[23]李金花,郑锋,刘秀玲.非线性反应扩散方程的广义条件对称[J].齐齐哈尔大学学报,2008,V24(5),68-69
[24]Zhang J. F., Dai C. Q., Yang Q.. Variable-coefficient F-expansion Method and Its Application to Nonlinear Schrodinger Equation[J]. Optics Communications,2005, V52(4-6):408-421
[25]Zhang S., Xia T. C.. A Generalized F-expansion Method with Symbolic Computat-ion Exactly Solving Broer-kaup Equations[J]. Applied Mathematics and Computat ion,2007, V189(1):836-843
[26]Zhang S., Xia T. C.. A Generalized F-expansion Method and New Exact Solution-s of Konopelchenko-Dubrovsky Equation[J]. Applied Mathematics and C-omputati on,2006, V183(2):1990-1200
[27]张金良,任东锋,王明亮.Davey-Strwartson的周期波解[J].数学物理学报,2005,V25(2)：213-219
[28]Zhang S. L., Lou S. Y., Qu C. Z.. New Variable Separation Approach:applicatio n to Nonlinear Diffusion Equations[J]. Phys. A,2003, V36:12223-12242
[29]Ablowitz M.. Method for Solving the sine-Gordon Equation[J]. Phys. Rev. Lett., 1973, V30:1262-1270
[30]Hirota R.. Exact Solutions of the Korteweg-de vries Equation for Multiple Collisi-ons of Solitions[J]. Phys. Rev. Lett.,1971, V27:1192-1196
[31]Wazwaz A.. The Hirota's Bilinear Method and the tanh-coth Method for Multiple-solitom Solutions of the Sawada-Kotera-Kadomtsev-Petviashvili Equation[J]. Appli ed Mathematics and Computation,2008, V200:160-166
[32]Wazwaz A.. Multiple-solitom Solutions for the KP Equation by Hirota's Bilinear Method and the tanh-coth Method[J]. Applied Mathematics and computation,200 7, V190:633-640
[33]Gabet L.. The Theoretical Foundation of the Adomian Method[J]. Computers and Mathematics with Applications,1994, V27(12):41-52
[34]Wang M.. Solitary Wave Solutions for Variant Boussinesq Equations[J]. Phys. Lett. A,1995, V99:169-172
[35]王明亮,李志斌,周宇斌.齐次平衡原则及其应用[J].兰州大学学报(自然科学),1999,V35(3)：8-19
[36]Fan E., Zhang H.. A Note on the Homogeneous Balance Method[J]. Phys. Lett. A,1998, V246:403-406
[37]Fan E.. Two New Applications of the Homogeneous Balance Method[J]. Phys. Le-tt. A,2000, V265(5):353-357
[38]闫振亚.复杂非线性波的构造性理论及其应用[M].北京：科学出版社,2006,28
[39]Wazwaz A.. The tanh Method for Traveling Wave Solutions of Nonlinear Equatio-ns[J]. Applied Mathematics and Computation,2004, V6:713-723
[40]Malfliet W.. The tanh Method:a Tool for Solving Certain Classes of Nonlinear Evolution and Wave Equations[J]. Computation and Applied Mathematics,2004 V164:529-541
[41]Malfiet W.. Solitary Wave Solutions of Nonlinear Wave Equations[J]. Phys. A,19 92, V60:650-654
[42]Wazwaz A.. The tanh Method:Exact Solutions of the sine-Gordon Equations[J]. Appl. Math. Comput.,2005, V167:1196-210
[43]Khalfallah M.. New Exact Traveling Wave Solutions of the (3+1) Dimensional K adomtsev-Petviashvili(KP) Equation[J]. Communications in Nonlinear Science and Namerical Simulation,2009, V14(4):1169-1175
[44]Parkes E.. An Automated tanh-function Method for Finding Solitary Wave Solutio-ns to Nonlinear Evolution Equations[J]. Computer Physics Communications,1996, V98:288-300.
[2]Liu S. K., Fu Z. T., Liu S. D.. Jacobi Elliptic Function Expansion Method and Pe riodic Wave Solutions of Nonlinear Wave Equations[J]. Physics Letters A,2001, V 28(9):69-74
[3]Filiz T., Ahmet B., Murat K.. Travelling Wave Solution of Nonlinear Evolution Eq uation by Using the First Integral Method[J]. Commun Nonlinear Sci Number Sim ular,2009, V14:1810-1815
[4]Feng Z. S.. The First Intergral Method to Study the Burgers-Korteweg-deVriesequat ion[J]. J. Phys. A,2002, V35:343-349
[5]Feng Z. S.. Exact Solution to an Approximate sine-Gordon Equation in (n+l)dime nsional Space[J]. Phys. Lett. A,2002, V302:64-76
[6]Feng Z. S., Wang X. H.. The First Intergral Method to the Two-dimensional Burg ers-Korteweg-deVriesequation[J]. Phys. Lett. A,2003, V308:173-178
[7]Naranmandula M., Wang K. X.. New Explicit Exact Solutions to a Nonlinear Disp ersive-dissipative Equation[J]. Chinese Physics,2004, V13(2):139-143
[8]Inc M.. New Exact Solutions for the ZK-MEW Equation by Using Symbolic Com-putation[J]. Appl. Math. Comput,2007, V189(508)
[9]He J., Wu X.. Exp-function Method for Nonlinear Wave Equations Chao Solutions and Fractals[J]. Phys. Lett. A,2006, V30(3):700-708
[10]Zhao M., Li C.. The Expansion Method Applied to Nonlinear Evolution Equation. http://www.paper.edu.cn/paper.php?serial number=200805-762
[11]祁新雷.若干非线性演化方程的精确解[D].西安：西北大学,2009
[12]史良马.变系数非线性方程的求解[D].安徽：安徽大学,2006
[13]Ablowitz M., Clarkson P.. Solutions Nonlinear Evolutions and Inverse Scattering Transform[M]. Cambridg University Press,1990
[14]Vakhnenko V., parkes E., Morrison A.. A Backiund Transformations and the Inver se Scattering Transform Method for the Generalized Vakhenko Equation Chaos sol ution Fract[J]. J. Phys. A,2003, V17(4):683-685
[15]Gardner C., Greene J., Kruskal M., etal. Method for Solving the Kortewege-de V ries Equation[J]. Phys. Lett.,1967, V19:1095-1907
[16]Hirota R.. A New Form of Backlund Transformations and Its Relation to the Inv erse Scattering Problem[J]. Prog. Theor. Phys.,1974, V52(5):1498-1512
[17]Lamb G.. Backlund Transformations for Certain Nonlinear Evolution Equations[J]. Math. phys.,1974, V15:2157-2165
[18]Wu H.. On Backlund Transformations for Nonlinear Partial Differential Equations [J].J. Math. Anal. Appl.,1995, V192:151-179
[19]Matveev V.. Darbous Transformations and Soliton Solutions[R]. Berlin,1991
[20]曹生让.几个非线性演化方程的精确解[D].江苏：江苏大学,2007
[21]李德生.若干非线性演化方程精确求解法的研究[D].大连：大连理工,2004
[22]Weiss J., Tabor M., Carnevale G. et al. The Painleve Property for Partica Differe-ntial Equations[J]. Math. Phys.,1983, V24:522-526
[23]李金花,郑锋,刘秀玲.非线性反应扩散方程的广义条件对称[J].齐齐哈尔大学学报,2008,V24(5),68-69
[24]Zhang J. F., Dai C. Q., Yang Q.. Variable-coefficient F-expansion Method and Its Application to Nonlinear Schrodinger Equation[J]. Optics Communications,2005, V52(4-6):408-421
[25]Zhang S., Xia T. C.. A Generalized F-expansion Method with Symbolic Computat-ion Exactly Solving Broer-kaup Equations[J]. Applied Mathematics and Computat ion,2007, V189(1):836-843
[26]Zhang S., Xia T. C.. A Generalized F-expansion Method and New Exact Solution-s of Konopelchenko-Dubrovsky Equation[J]. Applied Mathematics and C-omputati on,2006, V183(2):1990-1200
[27]张金良,任东锋,王明亮.Davey-Strwartson的周期波解[J].数学物理学报,2005,V25(2)：213-219
[28]Zhang S. L., Lou S. Y., Qu C. Z.. New Variable Separation Approach:applicatio n to Nonlinear Diffusion Equations[J]. Phys. A,2003, V36:12223-12242
[29]Ablowitz M.. Method for Solving the sine-Gordon Equation[J]. Phys. Rev. Lett., 1973, V30:1262-1270
[30]Hirota R.. Exact Solutions of the Korteweg-de vries Equation for Multiple Collisi-ons of Solitions[J]. Phys. Rev. Lett.,1971, V27:1192-1196
[31]Wazwaz A.. The Hirota's Bilinear Method and the tanh-coth Method for Multiple-solitom Solutions of the Sawada-Kotera-Kadomtsev-Petviashvili Equation[J]. Appli ed Mathematics and Computation,2008, V200:160-166
[32]Wazwaz A.. Multiple-solitom Solutions for the KP Equation by Hirota's Bilinear Method and the tanh-coth Method[J]. Applied Mathematics and computation,200 7, V190:633-640
[33]Gabet L.. The Theoretical Foundation of the Adomian Method[J]. Computers and Mathematics with Applications,1994, V27(12):41-52
[34]Wang M.. Solitary Wave Solutions for Variant Boussinesq Equations[J]. Phys. Lett. A,1995, V99:169-172
[35]王明亮,李志斌,周宇斌.齐次平衡原则及其应用[J].兰州大学学报(自然科学),1999,V35(3)：8-19
[36]Fan E., Zhang H.. A Note on the Homogeneous Balance Method[J]. Phys. Lett. A,1998, V246:403-406
[37]Fan E.. Two New Applications of the Homogeneous Balance Method[J]. Phys. Le-tt. A,2000, V265(5):353-357
[38]闫振亚.复杂非线性波的构造性理论及其应用[M].北京：科学出版社,2006,28
[39]Wazwaz A.. The tanh Method for Traveling Wave Solutions of Nonlinear Equatio-ns[J]. Applied Mathematics and Computation,2004, V6:713-723
[40]Malfliet W.. The tanh Method:a Tool for Solving Certain Classes of Nonlinear Evolution and Wave Equations[J]. Computation and Applied Mathematics,2004 V164:529-541
[41]Malfiet W.. Solitary Wave Solutions of Nonlinear Wave Equations[J]. Phys. A,19 92, V60:650-654
[42]Wazwaz A.. The tanh Method:Exact Solutions of the sine-Gordon Equations[J]. Appl. Math. Comput.,2005, V167:1196-210
[43]Khalfallah M.. New Exact Traveling Wave Solutions of the (3+1) Dimensional K adomtsev-Petviashvili(KP) Equation[J]. Communications in Nonlinear Science and Namerical Simulation,2009, V14(4):1169-1175
[44]Parkes E.. An Automated tanh-function Method for Finding Solitary Wave Solutio-ns to Nonlinear Evolution Equations[J]. Computer Physics Communications,1996, V98:288-300.