一类(3+1)维KdV方程的有理解及其怪波

汪春江,舒 级,李 倩,王云肖,杨 袁

(四川师范大学 数学与软件科学学院,四川 成都 610066)

一类(3+1)维KdV方程的有理解及其怪波

汪春江,舒 级*,李 倩,王云肖,杨 袁

(四川师范大学 数学与软件科学学院,四川 成都 610066)

讨论一类经典的(3+1)维KdV方程,该方程在流体动力学、等离子物理、气体动力学等方面有广泛应用.通过一个简单的符号计算方法得到方程的有理解,并讨论了在某些条件下的怪波解.

KdV方程; 精确解; 符号计算方法; 有理解; 怪波

一直以来,非线性现象都是基础数学和应用数学研究的主题,非线性演化方程的精确解研究在数学物理上有着重大作用,常系数方程[1-3]、变系数方程[4-5]、随机方程[6-7]的精确解已经被广泛研究,并产生了很多关于精确解的方法,如逆散映射法[8-9]、Backlünd变换[10-11]、达布变换[12]、齐次平衡法[13]、(G′/G)-展开法[14]等.

近年来,怪波已成为国内外研究的焦点.从怪波解的形式上看,通常是有理分式.在海洋学和其他学科领域,科学家们都发现了怪波现象[15-17],例如,Bose-Einstein凝聚物怪波事件[18-19],在等离子体的空间和表面的异常波[20-21],尤其是在光学领域[22-23],当光脉冲在光子晶体纤维中传输高能量时,怪波就会存在.对于非线性Schrödinger方程[24-25]的怪波与驻波、多怪波和高阶怪波[26-28]、明暗怪波解[29-30]已广泛被讨论.怪波不仅出现在深水中,浅水中也发现了怪波[31].从直观上看,怪波具有超常的波高,因此大多数学者和研究人员只能从波高角度对它进行定义,即认为波高大于有效波高2倍(或2.2倍)的单波可以称为怪波.在浅水中,怪波的产生取决于调制不稳定性:当波高kH<1.363 m[32],非线性聚焦过程停止.以水为介质的波不同于一般的阈值,浅水波高为kH<1/3 m[33].

在本文中,考虑(3+1)维KdV方程

(1)

其中u是关于x、y、z、t的函数,x、y、z、t是独立变量.它是物理学家和数学家感兴趣的方程之一.KdV方程的复合解[35]、变系数KdV方程的精确解[36]、高阶KdV方程的精确解[37-39]、新的KdV-mKdV方程的孤波[40]、广义的Hirota Satsuma耦合方程和偶合的MKdV方程[41-42]的孤立波解已经被研究.

本文的目的是通过一个简单的符号计算方法[43],构建出(3+1)维KdV的怪波和有理解.

1 一个简单的符号计算方法

一个简单的符号计算方法对于求解非线性偏微分方程是有效的.下面给出求解(3+1)维偏微分方程的主要步骤.

步骤 1 利用截断展开法[44-46],作代换

(2)

其中,α是一个常数,j1,j2,j3,j4≥1(j1,j2,j3,j4∈0,1,2,…).通过上述变换将一个(3+1)维的偏微分方程

(3)

转化为一个双线性方程

(4)

其中F是f,ft,fx,fy,fz,fxx,fxy,fxz,…的多项式.

步骤 2 假设f是一个关于x、y、z、t的2N阶多项式,给定

(5)

其中,系数ai,j,k,l(0≤i,j,k,l≤2N)是常数,满足

(6)

如果j1≥1.

步骤 3 把(5)式代入(4)式,并令关于xiyjzktl(0≤i+j+k+l≤2N)的系数为0,得到一系列的多项式方程.

步骤 4 利用Maple软件,可以算出系数aijkl,i,j,k,l∈(0,1,2,…).

步骤 5 把满足条件(6)的系数aijkl(i,j,k,l∈{0,1,2,…})代入(2)式之后得到非线性偏微分方程(3)的解.

2 有理解

下面将应用上述方法来求解(3+1)维KdV方程的解.根据截断展开法,用变换

(7)

将(3+1)维KdV方程变为

令f成为上述方程的一个解

(8)

为了构建方程(1)的有理解,假定f是一个2阶关于x、y、z、t的多项式

(9)

其中系数aijkl(0≤i,j,k,l≤2)是常数,满足

(10)

为了简单起见,假定方程(9),a2000=a0200=a0020=a0000=1,a0002=2,代入到方程(8)中,并令关于xiyjzktl(0≤i+j+k+l≤2)的系数为0,得到15个多项式方程:

为了后面描述的方便,令a0011=b,a1100=c,a1001=d,a0110=e,a0101=f,a1010=g,a0001=h,a0010=j,a0100=k,a1000=l,解方程组,得到如下解.

情形 1

f=-ge,

情形 2

f=-ge,

情形 3

f=-ge,

情形 4

f=-ge,

情形 5

情形 6

情形 7

情形 8

情形 9

情形 10

情形 11

情形 12

情形 13

情形1～13代入(7)式得到(3+1)维KdV方程的解,情形1～4有2个自由变量a1100、a0010.在这些情形中可以得到实值型的怪波解.情形5～12有一个自由变量a0010.在这些情况中得到复值型的有理解.然而,在情形13中,有一个自由变量a0010.在这个情形中,可以得到实值型的有理解.

1) 从情形1中,挑选出系数:

可以得到(3+1)维KdV方程的实值型怪波解

其中

(12)

2) 从情形5中,挑选出系数:

可以得到(3+1)维KdV方程的复值型有理解

(13)

其中

(14)

3) 从情形13中,挑选出系数:

可以得到(3+1)维KdV方程的实值型有理解

其中

(16)

3 结束语

本文运用符号计算方法得到了(3+1)维KdV方程的有理解和怪波解,这些怪波和有理解是非奇异的.这些解对理解怪波的产生机制有一定帮助.下一步将研究如何用简单的符号计算方法构造非线性演化方程的高阶怪波解.

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2010 MSC：35Q55

(编辑 周 俊)

Rogue Waves and Rational Solutions for a Class of (3+1)-dimensional KdV Equation

WANG Chunjiang,SHU Ji,LI Qian,WANG Yunxiao,YANG Yuan

(CollegeofMathematicsandSoftwareScience,SichuanNormalUniversity,Chengdu610066,Sichuan)

This paper discusses a classical (3+1)-dimensional KdV equation,which has broad applications in hydrodynamics,plasma physics,gas dynamics.We obtain rational solutions of this equation by a simple symbolic computation approach.Under some conditions,we find that some of rational solutions are rogue waves.

KdV equation; exact solution; symbolic computation approach; rational solution; rogue wave

2016-03-30

四川省科技厅应用基础计划项目(2016JY0204)和四川省教育厅自然科学重点基金(14ZA0031)

O175.27

A

1001-8395(2017)02-0157-06

10.3969/j.issn.1001-8395.2017.02.003

*通信作者简介：舒 级(1976—),男,教授,主要从事随机动力系统和偏微分方程的研究,E-mail:shuji2008@hotmail.com