非线性偏微分方程的行波解与分支
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摘要
本文用平面动力系统分支理论研究了广义Pochhammer-Chree方程的孤波与扭波解的分支，给出了方程在参数空间中所有分支集，以及孤波解与扭波解的个数，并给出了各种参数条件下的所有孤波与扭波解的精确公式。
研究了一类广义耦合非线性方程的行波解分支，得到了该方程的无穷多光滑与非光滑周期波的存在性，并在各种不同的参数条件下，给出了保证该方程上述解以及孤波解存在的充要条件。
Bifurcation of solitary waves and kink waves for Generalise Pochhammer-Chree Equation are studied, by using the bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown.Numbers of solitary waves and kink waves are given. Under various parameter conditions, all expicit formulars of solitary wave solutions and kink wave solutions are obtained.
By using the theory of planar dynamics system to a class of Coupling Nonlinear Equations,the existence of uncountably infinite many smooth and non-smooth periodic wave solutions and solitary wave solutions is obtained. Under different parametric conditions, various sufficient conditions to gurantee the existence of the above solutions are given.
研究了一类广义耦合非线性方程的行波解分支，得到了该方程的无穷多光滑与非光滑周期波的存在性，并在各种不同的参数条件下，给出了保证该方程上述解以及孤波解存在的充要条件。
Bifurcation of solitary waves and kink waves for Generalise Pochhammer-Chree Equation are studied, by using the bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown.Numbers of solitary waves and kink waves are given. Under various parameter conditions, all expicit formulars of solitary wave solutions and kink wave solutions are obtained.
By using the theory of planar dynamics system to a class of Coupling Nonlinear Equations,the existence of uncountably infinite many smooth and non-smooth periodic wave solutions and solitary wave solutions is obtained. Under different parametric conditions, various sufficient conditions to gurantee the existence of the above solutions are given.
引文
[1] LL.Bogolubsky , Some examples of inelastic soliton interaction , Computer Physics Communications , 13 (1977) , 2: 149-155．
[2] P.A.Clarcson, R.J.LeVeque and R.Saxton, Solinary wave interactions in elasticrods, Studies in Applied Mathematics, 75 (1986), 1: 95-122.
[3] S.N.Chow and J. J.K.Hale, Methods of bifurcation theory, Springer-Verlay, New York, 1982.
[4] J.Guckenheimer and P. Holmes, Dynamical systems and bifurcations of vector fields, Springer-Verlay, New York, 1982.
[5] Jibin Li and Zhengrong Liu, Smooth and non-smooth travelling waves in a nonlinearly dispersive equation, Appl.Math. Modelling, 25(2000), 41-56.
[6] R. Saxton, Existence of silutions for a finite nonlinearly hyperelestic rod J. Math.Anal.Appl., 105 (1985), 1:59-75.
[7] Weiguo Zhang and Wenxiu Ma, Explict solitary wave solutions to generalized Pochammer-Chree equations, Applied Mathematics and Mechanics, 20 (1999), 6:666-674.
[8] 范恩贵、张鸿庆，非线性波动方程的孤波解，物理学报，46(1997)7：1254-1258。
[9] 夏铁成、张鸿庆、闫振亚，一类非线性演化方程新的显式行波解，应用数学与力学，22(2001)7：701-705。
[10] 闫振亚、张鸿庆、范恩贵，一类非线性演化方程新的显式行波解，物理学报，48(1995)1：1-5。
[11] Guo Boling and Tan Shaobin ,Global smooth solution for a coupled nonlinear wave equations, [J]Math Meth in the Appi Sci ,1991,14:419—425.
[12] Jibin Li and Lijun Zhang ,Bifurcations of Traveling Wave Solutions in Generalized Pochhammer-Chree Equation. (preprint)
[13] 李继彬等，《非线性微分方程》，成都科技大学出版社，(1998)。
[14] 张芷芬等，《微分方程定性理论》，科学出版社，中国．北京(1985)。
[2] P.A.Clarcson, R.J.LeVeque and R.Saxton, Solinary wave interactions in elasticrods, Studies in Applied Mathematics, 75 (1986), 1: 95-122.
[3] S.N.Chow and J. J.K.Hale, Methods of bifurcation theory, Springer-Verlay, New York, 1982.
[4] J.Guckenheimer and P. Holmes, Dynamical systems and bifurcations of vector fields, Springer-Verlay, New York, 1982.
[5] Jibin Li and Zhengrong Liu, Smooth and non-smooth travelling waves in a nonlinearly dispersive equation, Appl.Math. Modelling, 25(2000), 41-56.
[6] R. Saxton, Existence of silutions for a finite nonlinearly hyperelestic rod J. Math.Anal.Appl., 105 (1985), 1:59-75.
[7] Weiguo Zhang and Wenxiu Ma, Explict solitary wave solutions to generalized Pochammer-Chree equations, Applied Mathematics and Mechanics, 20 (1999), 6:666-674.
[8] 范恩贵、张鸿庆，非线性波动方程的孤波解，物理学报，46(1997)7：1254-1258。
[9] 夏铁成、张鸿庆、闫振亚，一类非线性演化方程新的显式行波解，应用数学与力学，22(2001)7：701-705。
[10] 闫振亚、张鸿庆、范恩贵，一类非线性演化方程新的显式行波解，物理学报，48(1995)1：1-5。
[11] Guo Boling and Tan Shaobin ,Global smooth solution for a coupled nonlinear wave equations, [J]Math Meth in the Appi Sci ,1991,14:419—425.
[12] Jibin Li and Lijun Zhang ,Bifurcations of Traveling Wave Solutions in Generalized Pochhammer-Chree Equation. (preprint)
[13] 李继彬等，《非线性微分方程》，成都科技大学出版社，(1998)。
[14] 张芷芬等，《微分方程定性理论》，科学出版社，中国．北京(1985)。