一個(gè)組合方程的單孤子解和周期尖波解

楊海霞

(1.西北大學(xué)數(shù)學(xué)系,陜西西安 710069;2.西北大學(xué)非線(xiàn)性研究中心,陜西西安 710069)

一個(gè)組合方程的單孤子解和周期尖波解

楊海霞1,2

(1.西北大學(xué)數(shù)學(xué)系,陜西西安 710069;2.西北大學(xué)非線(xiàn)性研究中心,陜西西安 710069)

構(gòu)造一個(gè)組合方程的單孤子解和周期尖波解.應(yīng)用格林函數(shù)的性質(zhì),以及求一個(gè)非線(xiàn)性偏微分方程(簡(jiǎn)稱(chēng)PDE)弱解的方法.求出了這個(gè)組合方程的單孤子解和周期尖波解,推廣了前人的研究成果.

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

1 引言

在近幾十年來(lái),很多學(xué)者對(duì)Camassa-Holm(簡(jiǎn)稱(chēng)CH)方程,modified Camassa-Holm(簡(jiǎn)稱(chēng)mCH)方程和Novikov方程進(jìn)行了深入研究,通過(guò)求出它們的單孤子解,多孤子解和周期尖波解,從而對(duì)它們的性質(zhì)進(jìn)行了研究,比如它的拉克斯對(duì),雙哈密頓結(jié)構(gòu),進(jìn)而得到了它們的可積性.還有像解的爆破性,穩(wěn)定性問(wèn)題,從而可以看出,要研究一個(gè)新的方程,方程的解的問(wèn)題占據(jù)著重要意義.

已經(jīng)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,猜測(cè)這個(gè)組合方程也應(yīng)該有形如這樣的解,通過(guò)運(yùn)用求弱解的方法[1,8-9],證實(shí)了猜想,并且求出了這個(gè)組合方程的單孤子解和周期尖波解.

2 組合方程的孤立波解

3 組合方程的周期尖波解

4 結(jié)論

通過(guò)運(yùn)用求弱解的方法以及格林函數(shù)的性質(zhì),求出了一個(gè)新的組合方程的單孤子解和周期尖波解,這對(duì)于以后進(jìn)一步研究它的其它性質(zhì)(比如它的拉克斯對(duì),雙哈密頓結(jié)構(gòu)等)都有著不可或缺的作用.

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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.

楊海霞(1988-),碩士生,研究方向:非線(xiàn)性偏微分方程

2010 MSC:35J15