非线性微分方程的行波解分支
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摘要
本文应用动力系统分支理论对一类非线性分数幂方程的动力学行为进行研究,给出了不同参数空间的行波解相图、分支集以及孤立波和扭子波解的存在条件,并得出部分孤立波和扭子波解的精确公式。同时还应用动力系统分支理论对一类修正Camassa-Holm方程进行研究,在参数空间中给定的区域内获得了系统在各种参数条件下可能存在的孤立波解、不可数无穷多光滑和非光滑周期行波解的存在性条件。
In this paper, the bifurcation behavior for a class of nonlinear evolution equations is studied by using the method of the bifurcation theory of dynamical systems. The bifurcation sets and phase portraits of the travelling wave equation in the different regions of the parameter space are given. The different parameter conditions for the existence of solitary wave solutions and kink wave solutions are rigorously determined. Explicit formulas of solitary wave solutions and kink wave solutions are given partly. At the same time, by employing the theory of bifurcations of dynamical systems to a modified Camassa-Holm equation, the existence of solitary wave solutions, uncountably infinite many smooth and non-smooth periodic traveling wave solutions, is obtained in the given areas of parametric space.
In this paper, the bifurcation behavior for a class of nonlinear evolution equations is studied by using the method of the bifurcation theory of dynamical systems. The bifurcation sets and phase portraits of the travelling wave equation in the different regions of the parameter space are given. The different parameter conditions for the existence of solitary wave solutions and kink wave solutions are rigorously determined. Explicit formulas of solitary wave solutions and kink wave solutions are given partly. At the same time, by employing the theory of bifurcations of dynamical systems to a modified Camassa-Holm equation, the existence of solitary wave solutions, uncountably infinite many smooth and non-smooth periodic traveling wave solutions, is obtained in the given areas of parametric space.
引文
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[5] A.M. Wazwaz. New compact and noncompact solutions for two variant of a modified Camassa-Holm equation[J]. Applied Mathematics and Computation, 2005, 163: 1165-1179.
[6] L. Tian, X. Song. New peaked solitary wave solutions of the gerealized Camassa-Holm equation[J]. Chaos, Solitions and Fractals, 2004, 19: 621-637.
[7] J.P. Boyd. Peakons and cashiodal waves: Travelling wave solutions of the Camassa-Holm equation[J]. Applied Mathematics and Computation, 1997, 81(2-3): 173-187.
[8] JibinLi and Lijun Zhang. Bifurcation of traveling solutions in generalized Pochhammer-Chree equation[J]. Chaos, Solitions and Fractals, 2002,14: 581-593.
[9] Jibin Li, Hong Li and Shumin Li. Bifurcations of travelling wave solutions for the generalized Kadomtsev-Petviashili eqution.
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[2] Yasunori Nejoh, New spliky waves, explosive modes and periodic progoressive waves in a magnetised plasma[J], J. Phys. A: Math. Gen. 1990, 23: 1973-1984.
[3] Li Jibin and Liu Zhengrong. Smooth and non-smooth traveling waves in a nonlinearly dispersive equation[J]. Appl. Math. Modelling, 2000, 25: 41-56.
[4] Li Jibin and Liu Zhengrong. Traveling wave solutions for a class of nonlinear dispersive equations[J]. Chinese Annals of Mathematics, Ser B, 2002, 23(3): 397-418.
[5] A.M. Wazwaz. New compact and noncompact solutions for two variant of a modified Camassa-Holm equation[J]. Applied Mathematics and Computation, 2005, 163: 1165-1179.
[6] L. Tian, X. Song. New peaked solitary wave solutions of the gerealized Camassa-Holm equation[J]. Chaos, Solitions and Fractals, 2004, 19: 621-637.
[7] J.P. Boyd. Peakons and cashiodal waves: Travelling wave solutions of the Camassa-Holm equation[J]. Applied Mathematics and Computation, 1997, 81(2-3): 173-187.
[8] JibinLi and Lijun Zhang. Bifurcation of traveling solutions in generalized Pochhammer-Chree equation[J]. Chaos, Solitions and Fractals, 2002,14: 581-593.
[9] Jibin Li, Hong Li and Shumin Li. Bifurcations of travelling wave solutions for the generalized Kadomtsev-Petviashili eqution.
[10] 李继彬.非线性微分方程[M].昆明:云南科技出版社,1987.
[11] 张锦炎,冯贝叶.常微分方程几何理论与分支问题(第二版)[M].北京:北京大学出版社,1980.
[12] 黎捷.Maple 9.0符号处理及应用[M].北京:科学出版社,2004.
[13] 李继彬,赵晓华,刘正荣.广义哈密顿系统理论及其应用[M].北京:科学出版社,1994.
[14] 张芷芬,丁同仁,黄文灶等.微分方程定性理论[M].北京:科学出版社,2003.
[15] 徐桂芳.积分表[M].上海:上海科学技术出版社,1962.
[16] 王高雄,周之铭等.常微分方程[M].北京:高等教育出版社,1996.