具任意次非线性项的Lienard方程的精确解及其应用

摘要/Abstract

摘要：

该文推导了具任意次非线性项的Liénard方程a″(ξ)+la(ξ)+ma\+q(ξ)+na\+\{2q-1\}(ξ)=0和\{a″(ξ)\}+ra′(ξ)+la(ξ)+ma\+q(ξ)+na\+\{2q-1\}(ξ)=0解的若干性质，通过适当变换，并结合假设待定法求出了它们的钟状和扭状显式精确解.据此，求出了一批具任意次非线性项的发展方程的钟状和扭状显式精确孤波解，其中包括广义BBM型方程、二维广义Klein Gordon方程、广义Pochhammer Chree方程和非线性波方程等.

关键词: 孤波;Liénard方程, 非线性发展方程, 精确解, 待定假设法.

Abstract:

In this paper, the authors first establish some properties of solutions for Liénard equation with nonlinear terms of any order. Then, explicit exact solutions for the Liénard equation are obtained by proper transformation and undetermined assumption method. By means of these solutions, the authors obtain explicit exact bell and kink profile solitary wave solutions for many nonlinear evolution equation with nonlinear terms of any degree. These nonlinear equations include generalized BBM type, generalized KleinGordon, generalized Pochhammer Chree and generalized wave equation.

Key words: Solitary wave； , Liénard equation； Nonlinear evolution equation exact solution； Undetermined assumption method.

中图分类号:

35Q20

张卫国, 常谦顺, 李永声. 具任意次非线性项的Lienard方程的精确解及其应用[J]. 数学物理学报, 2005, 25(1): 119-129.

ZHANG Wei-Guo, CHANG Qian-Shun, LI Yong-Qing-. Explicit Exact Solutions for Lienard Equation with Nonlinear Terms of any Degree and its Applications[J]. Acta mathematica scientia,Series A, 2005, 25(1): 119-129.

参考文献

[1]Villari G. On the qualitative behaviour of solutions of Liénard equation. J Diff Equ, 1987, 67(2): 269-277

[2]Dumortier F, Rousseau C. Cubic Liénard equations with linear damping. Nonlinearity, 1990，3: 1015-1039

[3]韩茂安. 一类广义Liénard方程的有界性. 科学通报，1995,40 (21): 1925-1928

[4]黄立宏，庾建设. 广义Liénard方程非平凡周期解的存在性. 应用数学，1995,8(2): 172-176

[5]Benjamin T B, Bona J L, Mahong J J. Model equations for long waves in no nlinear dispersive systems. Philos Trans R Soc(Ser A), 1972, 272: 47-78

[6]Medeiros L A, Perla G M. Existance and uniqueness for periodic solutions of the BenjaminBonaMahony equation. SIAM J Math Anal, 1977, 8(5): 792-799

[7]张卫国，王明亮. BBBM方程的一类准确行波解及结构. 数学物理学报，1992，12(3)：325-331

[8]Ablowitz M J. Lectures on the inverse scattering transform. Studies in Applied Mathematics,1978,58(11):17-94

[9]Whitham G B. Linear and Nonlinear Waves. New York: John Wiley, 1974 

[10]Bogolubsky I L. Some examples of inelastic soliton interaction. Computer Physics Communications, 1977, 13(2): 149-155

[11]Clarkson P A, Le Veque R J, Saxton R. Solitarywave interactions in elastic rods. Studies in Applied Mathematics, 1986,75(1): 95-122

[12]JP4]Saxton R. Existence of solutions for a finite nonlinearly hyperelastic rod. J Math Anal Appl,1985,105(1):59-75

[13]Wadati M. Wave propagation in nonlinear lattice (Ⅰ), (Ⅱ). J Phys Soc Japan, 1975, 38(3): 673-680

[14]Dodd R K, et al. Solitons and Nonlinear Wave Equations. London: AcademicPress, 1982

[15]Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. London: Cambridge University Press, 1991

[16]Farlow S J. Partial Differential Equation for Scientists and Engineers. New York: Wiley Interscience, 1982

[17]Kamke E著，张鸿林译. 常微分方程手册. 北京：科学出版社，1980

[18]张卫国. 几类具5次强非线性项的发展方程的显式精确孤波解. 应用数学学报，1998, 21(2): 249-256

[19]范恩贵，张鸿庆. 非线性波动方程的孤波解. 物理学报，1997,46(7): 1254-1258