一类非线性偏微分方程的多孤子解

李 伟，张智欣

(渤海大学 数理学院，辽宁 锦州 121013)

一类非线性偏微分方程的多孤子解

李 伟，张智欣

(渤海大学 数理学院，辽宁 锦州 121013)

许多重要的自然科学问题和工程问题都可以归结为非线性偏微分方程。从传统的角度来看,非线性偏微分方程的多孤子解是很难得到的。经过几十年的研究和探索,已经发现了一些构造精确解的方法。借助于科尔-霍普夫变换和Af+B=0方法，获得了Burgers方程和KP方程的多孤子解。该方法能够解决一系列偏微分方程。

科尔-霍普夫变换；Af+B=0方法；多孤子解

1 Introduction

Soliton is an important feature of nonlinearity and can be found in many scientific applications.Many systematic methods are used for studying the nonlinear evolution equations that give rise to solitons.The inverse scattering method, the Backlund transformation method, the Darboux transformation method, the Hirota bilinear method[1-5], and the Hereman-Nuseir method[6] are the most commonly used methods.The Hirota’s bilinear method is rather heuristic and possesses significant features that make it practical for the determination of multiple soliton solutions[7-13].developed a modified form of the Hirota’s method that facilitates the computational work.The computer symbolic systems such as Maple, Mathematica can be used to overcome the tedious calculations.

In this work, we will examine two kinds of equations that play a significant role in this field.The Burgers equation, theKPequation that will be examined, reads

(1)

(2)

In this work we will employ the Cole-Hopf transformation method and theAf+B=0 method to handle these two equations.We aim to obtain multiple-kink solutions for each equation.

2 The Af+B=0 method

We will consider the NPED

(3)

Our specific practice:

We first use the Cole-Hopf transformation

(4)

that will carry (3) to

Af+B=0

(5)

whereA,Bare the functions offx,ft,fxx,ftt,fxt, …, do not containf.

We set up

(6)

then the solution of (6) is solution of(5).

1) for single solution, we use

(7)

Substituting (7) into (6) and solvingr1, we findr1=r1(p1).

2) for two-solition solutions, we use

(8)

Substituting (8) into (5) and solvinga12, we finda12=a12(p1,p2).

3) for three-solition solutions, we use

(9)

Substituting (9) into (5) and solvingb123, we findb123=b123(p1,p2,p3).

3 The Burgers equation

We first use the Cole-Hopf transformation

that will carry (1) to

(11)

We set up

(12)

For single solution, we use

(13)

Substituting (13) into (12) and solving r1, we find

(14)

wherep1is arbitrary constant.

Substituting (13) and (14) into (10) give the single-kink solution

(15)

Therefore, we assume that the two-kink solutions for

(16)

wherep1,p2are arbitrary constants.

Substituting (16) into (11) and solvinga12, we find

(17)

Substituting (16) and (17) into (10) give the two-kink solutions

(18)

For three-solition solutions, we set

(19)

Substituting (19) into (11) and solvingb123, we find

(20)

Substituting (19) and (20) into (10) give the three-kink solutions

(21)

4 The KP equation

We use the Cole-Hopf transformation

(22)

that will carry the KP equation(2) to

(23)

We set up

(24)

For single solution, we use

(25)

Substituting (25) into (24) and solvingr1, we find

(26)

wherep1,q1are arbitrary constants.

Substituting (25) and (26) into (22) give the single-kink solution

(27)

Therefore, we assume that the two-kink solutions for

(28)

wherep1,p2are arbitrary constants.

Substituting (28) into (23) and solvinga12, we find

(29)

Substituting (28) and (29) into (22) give the two-kink solutions

(30)

wherep1,p2are arbitrary constants.

For three-solition solutions, we set

(31)

Substituting (31) into (23) and solvingb123, we find

(32)

Substituting (31) and (32) into (22) give the three-kink solutions

(33)

wherepi(i=1, 2, 3) are arbitrary constants.

5 Discussions

Two models, the Burgers equation, and theKPequation are studied.Multiple-kink solutions are formally derived for each equation.The results obstained we generalized to some equation.

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(责任编辑 陈 艳)

N-Soliton Solutions for a Class of Nonlinear Partial Differential Equations

LI Wei，ZHANG Zhi-xin

(College of Mathematical, Bohai University, Jinzhou 121013, China)

Many significant natural science and engineering problems can be attributed to nonlinear partial differential equation. From the traditional point of view, n-soliton solutions of partial differential equation are hard to get. After several decades of research and exploration, we have found some tectonic exact solution method. With the help of Cole-Hopf transformation method and theAf+B=0 method, n-soliton solutions of the Burgers equation and the KP equation were presented. This method could solve a series of partial differential equations.

the Cole-Hopf transformation;Af+B=0 method; multiple-soliton solution

2016-11-24 基金项目：国家自然科学基金资助项目(11547005)

李伟(1977—),男， 辽宁锦州人，硕士，主要从事偏微分方程研究， E-mail:1344462965@qq.com。

李伟，张智欣.一类非线性偏微分方程的多孤子解[J].重庆理工大学学报(自然科学)，2017(3):171-174.

format：LI Wei，ZHANG Zhi-xin.N-Soliton Solutions for a Class of Nonlinear Partial Differential Equations[J].Journal of Chongqing University of Technology(Natural Science)，2017(3):171-174.

10.3969/j.issn.1674-8425(z).2017.03.026

O175.2

A

1674-8425(2017)03-0171-04