一个组合方程的单孤子解和周期尖波解

杨海霞

(1.西北大学数学系,陕西西安 710069;2.西北大学非线性研究中心,陕西西安 710069)

一个组合方程的单孤子解和周期尖波解

杨海霞1,2

(1.西北大学数学系,陕西西安 710069;2.西北大学非线性研究中心,陕西西安 710069)

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.

Camassa-Holm方程;modified Camassa-Holm方程;Novikov方程;孤子解;周期尖波解

1 引言

在近几十年来,很多学者对Camassa-Holm(简称CH)方程,modified Camassa-Holm(简称mCH)方程和Novikov方程进行了深入研究,通过求出它们的单孤子解,多孤子解和周期尖波解,从而对它们的性质进行了研究,比如它的拉克斯对,双哈密顿结构,进而得到了它们的可积性.还有像解的爆破性,稳定性问题,从而可以看出,要研究一个新的方程,方程的解的问题占据着重要意义.

已经知道CH方程有孤子解[1-4]u=ae-|x-ct|,(c＞0)和周期尖波解[5-8]

的周期尖波解.于是,猜测这个组合方程也应该有形如这样的解,通过运用求弱解的方法[1,8-9],证实了猜想,并且求出了这个组合方程的单孤子解和周期尖波解.

2 组合方程的孤立波解

3 组合方程的周期尖波解

4 结论

通过运用求弱解的方法以及格林函数的性质,求出了一个新的组合方程的单孤子解和周期尖波解,这对于以后进一步研究它的其它性质(比如它的拉克斯对,双哈密顿结构等)都有着不可或缺的作用.

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Peakons and periodic cusp wave solutions of a combination equation

Yang Haixia1,2

(1.Department of Mathematics,Northwest University,Xi′an 710127,China; 2.Center for Nonlinear Studies,Northwest University,Xi′an 710069,China)

In order to study the single-soliton solutions and periodic peakons of a combination equation.By the application of the property of Green′s function,as well as seeking a PDE weak solution approach.The singlesoliton solutions and periodic peakons of the combination equation are obtained.The single-soliton solutions and periodic peakons of the combination equation are constructed,which generalizes the results of previous studies.

Camassa-Holm equation,modified Camassa-Holm equation,Novikov equation, peakons,period peakons

O175.29

A

1008-5513(2013)03-0306-12

10.3969/j.issn.1008-5513.2013.03.013

2013-04-03.

杨海霞(1988-),硕士生,研究方向:非线性偏微分方程

2010 MSC:35J15