一类非线性偏微分方程的n-孤子解
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摘要
微分方程包含线性和非线性微分方程。微分方程研究的主体是非线性微分方程,特别是非线性偏微分方程。很多意义重大的自然科学和工程技术问题都可归结为非线性偏微分方程的研究。另外,随着研究的深入,有些原来可用线性偏微分方程近似处理的问题,也必须考虑非线性的影响。从传统的观点来看,求偏微分方程的解是十分困难的。经过几十年的研究和探索,人们已经找到了一些构造解的方法。借助Cole-Hope变换,A=0且B=0为Af+B=0的解,获得了(2+1)维Burgers方程和Kdv方程的n-孤子解。这种方法可以求解一系列的偏微分方程。
Differential equations contain linear and nonlinear differential equations. Research of the nonlinear differential equations are the subject of differential equations, especially nonlinear partial differential equations. Many significant natural science and engineering problems can be attributed to nonlinear partial differential equation. In addition, With the development of research, some of the original with linear partial differential equation approximation problem must also consider nonlinear effects. From the traditional point of view, the solutions of partial differential equation is very difficult. After several decades of research and exploration, we have found some tectonic solution method. In this paper, With the help of Cole-Hope transform, one of the conditions for the equation Af+B=0 to be true if A=0 and B=0, n-soliton solutions of(2+1) dimensional Burgers equation and Kdv equation have been presented. This method could solve a series of partial differential equations.
Differential equations contain linear and nonlinear differential equations. Research of the nonlinear differential equations are the subject of differential equations, especially nonlinear partial differential equations. Many significant natural science and engineering problems can be attributed to nonlinear partial differential equation. In addition, With the development of research, some of the original with linear partial differential equation approximation problem must also consider nonlinear effects. From the traditional point of view, the solutions of partial differential equation is very difficult. After several decades of research and exploration, we have found some tectonic solution method. In this paper, With the help of Cole-Hope transform, one of the conditions for the equation Af+B=0 to be true if A=0 and B=0, n-soliton solutions of(2+1) dimensional Burgers equation and Kdv equation have been presented. This method could solve a series of partial differential equations.
引文
[ 1 ]李德生,张鸿庆.一类高维耦合的非线性演化方程的简单求解[J].物理学报,2004,53(6):1636-1638.
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[ 3 ]李翊神.孤子与可积系统[M].上海:上海科技教育出版社,1999:45-65.
[ 4 ]陈登远.孤子引论[M].北京:科学出版社,2006:14-44.
[ 5 ]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及几何应用[M].上海:上海科学技术出版社,1999:3-35.
[ 6 ]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法[J].物理学报,1998,47(3):356-362.
[ 7 ]桑波,伊继金,刘健文.常系数线性微分方程组的解矩阵[J].沈阳师范大学学报(自然科学版),2010,28(3):343-346.
[ 8 ]HEREMAN W,NUSEIR A.Symbolic methods to construct exact solutions of nonlinear partial differential equations[J].Math Comput Simul,1997,43:13-27.
[ 9 ]MATSUNO Y.Bilinear Transformation Method[M].Beijing:Academic Press,1984.
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[12]WAZWAZ A M.Multiple soliton solutions for the (2+1)-dimensional asymmetric Nizhanik-Novikov-Veselov equation nonlinear Ana[J].Theory Meth Appl,2010,72:1314-1318.
[13]WAZWAZ A M.The (2+1) and (3+1)-dimensional CBS equation:multiple soliton solutions and multiple singular soliton solutions[J].Multiple,Zeitschrift fur Naturforschung A(ZNA),2010,65a:173-181.
[14]WAZWAZ A M.The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves[J].Appl Math Comput,2008,201:489-503.
[15]WAZWAZ A M.Solitary wave solutions of the the generalized shallow water wave (GSWW) equation by Hirota's method,tanh-coth method and expfunction method[J].Appl Math Comput,2008,202:275-286.
[16]李志斌.非线性数学物理方程的行波解[M].北京:科技出版社,2006.
[ 2 ]李志斌.非线性数学物理方程的行波解[M].北京:科技出版社,2006:57-92.
[ 3 ]李翊神.孤子与可积系统[M].上海:上海科技教育出版社,1999:45-65.
[ 4 ]陈登远.孤子引论[M].北京:科学出版社,2006:14-44.
[ 5 ]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及几何应用[M].上海:上海科学技术出版社,1999:3-35.
[ 6 ]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法[J].物理学报,1998,47(3):356-362.
[ 7 ]桑波,伊继金,刘健文.常系数线性微分方程组的解矩阵[J].沈阳师范大学学报(自然科学版),2010,28(3):343-346.
[ 8 ]HEREMAN W,NUSEIR A.Symbolic methods to construct exact solutions of nonlinear partial differential equations[J].Math Comput Simul,1997,43:13-27.
[ 9 ]MATSUNO Y.Bilinear Transformation Method[M].Beijing:Academic Press,1984.
[10]WAZWAZ A M.Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations[J].Phys Scripta,2010,81:035005.
[11]WAZWAZ A M.Multiple soliton solutions for coupled KdV and coupled KP systems[J].Can J Phys,2010,87(12):1227-1232.
[12]WAZWAZ A M.Multiple soliton solutions for the (2+1)-dimensional asymmetric Nizhanik-Novikov-Veselov equation nonlinear Ana[J].Theory Meth Appl,2010,72:1314-1318.
[13]WAZWAZ A M.The (2+1) and (3+1)-dimensional CBS equation:multiple soliton solutions and multiple singular soliton solutions[J].Multiple,Zeitschrift fur Naturforschung A(ZNA),2010,65a:173-181.
[14]WAZWAZ A M.The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves[J].Appl Math Comput,2008,201:489-503.
[15]WAZWAZ A M.Solitary wave solutions of the the generalized shallow water wave (GSWW) equation by Hirota's method,tanh-coth method and expfunction method[J].Appl Math Comput,2008,202:275-286.
[16]李志斌.非线性数学物理方程的行波解[M].北京:科技出版社,2006.