一类耦合方程的单孤子解

李壹宏

(西北大学数学系,陕西西安 710127)

一类耦合方程的单孤子解

李壹宏

(西北大学数学系,陕西西安 710127)

利用检验函数定义弱解的方法来求解含有任意常数k1,k2的目标方程的单孤子解.给出了目标方程的单孤子解与任意常数k1,k2的关系.

修正的Camassa-Holm方程;Novikov方程;单孤子解

1 引言

随着对非线性问题的不断探究,现阶段最为关注的方程之一就是Camassa-Holm方程[1].本文主要研究与Camassa-Holm方程有关的一类方程,形式如下:

此方程是由Novikov在研究含有平方或立方非线性方程的非局部对称分类中获得[2].同时文献[2]中证明方程(2)有Lax对,故此方程是可积的.文献[3]表明此方程有双Hamilton结构和无穷多守恒律.方程(2)有如下的单孤子解[4]:

2 单孤子解求解

图1 当k1=1,k2=1,波速为c=1时的单孤子解

[1]Camassa R,Holm D D.A integrable shallow water equation with peaked soliton[J].Phys.Rev.Lett., 1993,71(11):1661-1664.

[2]Novikov V.Generalizations of the Camassa-Holm equation[J].Phys.A,2009,42(34):14-24.

[3]Hone A,Wang J P.Integrable peakon equations with cubic nonlinearity[J].Phys.A,2008,41:110-120.

[4]Hone A N,Lundmark H,Szmigielski J.Explicit multipeakon solutions of Novikov′s cubically nonlinear integrable Camassa-Holm type equation[J].Dyn.Partial Differ.Equ.,2009,6:253-289.

[5]Fuchssteiner B.Some tricks from the symmetry-toolbox for nonlinear equations:generalizations of the Camassa-Holm equation[J].Pysica D,1996,95:229-243.

[6]Olver P G,Rosennau P.Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support[J].Phys.Rev.E,1996,53:1900-1906.

[7]Gui Guilong,Liu Y,Olver P J,et al.Wave-breaking and peakons for a modified Camassa-Holm equation[J]. Commun.Math.Phys.,2013,319:731-759.

Single peakons for a combined equation

Li Yihong

(Department of Mathematics,Northwest University,Xi′an710127,China)

Using the way to define the weak solution by test functions,we obtain the peakon soliton of a system with cubic nonlinearity,which has arbitrary constants k1,k2.Finally we get the relationship between the peakon soliton and k1,k2.

modified Camassa-Holm equation,Novikov eqaution,peakon soliton

O175.29

A

1008-5513(2013)03-0287-06

10.3969/j.issn.1008-5513.2013.03.010

2013-03-15.

李壹宏(1988-),硕士生,研究方向：非线性偏微分方程.

2010 MSC:35J15