﻿ 有理函数变换法求解两类扩展的(3 + 1)维Jimbo-Miwa方程 The Transformed Rational Method for Solving Two Extended (3 + 1)-Dimensional Jimbo-Miwa Equations

Vol. 07  No. 10 ( 2018 ), Article ID: 27212 , 9 pages
10.12677/AAM.2018.710145

The Transformed Rational Method for Solving Two Extended (3 + 1)-Dimensional Jimbo-Miwa Equations

Jinting Ha1, Xue Guan2, Huiqun Zhang1

1Department of Mathematics and Statistics, Qingdao University, Qingdao Shandong

2School of Science, Beijing University of Posts and Telecommunications, Beijing

Received: Oct. 1st, 2018; accepted: Oct. 15th, 2018; published: Oct. 22nd, 2018

ABSTRACT

In this paper, two extended (3 + 1)-dimensional Jimbo-Miwa equations can be researched with the transformed rational function method. As a result, new exact travelling wave solutions for two extended (3 + 1)-dimensional Jimbo-Miwa equations are obtained by means of Maple software. What’s more, if we choose different ordinary differential equations, we can obtain different types of travelling wave solutions which supplement the existing literatures.

Keywords:(3 + 1)-Dimensional Jimbo-Miwa Equation, Transformed Rational Function Method, New Exact Travelling Wave Solutions

1青岛大学，数学与统计学院，山东 青岛

2北京邮电大学理学院，北京

1. 引言

KP可积族中的第二个方程为(3 + 1)维JM方程 [13] ，

${u}_{xxxy}+3{u}_{y}{u}_{xx}+3{u}_{x}{u}_{xy}+2{u}_{yt}-3{u}_{xz}=0,$ (1)

(3 + 1)维JM方程(1)已经被许多方法所研究，比如扩展的有理函数变换法 [14] ，扩展的Tanh方法 [15] ，Hirtoa双线性法 [16] [17] ，多指数函数法 [18] ，有理函数变换法 [12] ，Bell多项式法 [19] 等。由于(3 + 1)维JM方程好的可积性质，它可以被扩展成如下的方程，

${u}_{xxxy}+3{u}_{y}{u}_{xx}+3{u}_{x}{u}_{xy}+2{u}_{yt}-3\left({u}_{xz}+{u}_{yz}+{u}_{zz}\right)=0,$ (2)

${u}_{xxxy}+3{u}_{y}{u}_{xx}+3{u}_{x}{u}_{xy}+2\left({u}_{yt}+{u}_{xt}+{u}_{zt}\right)-3{u}_{xz}=0.$ (3)

2. 有理函数变换法

$P\left(x,t,{u}_{t},{u}_{x},{u}_{xx},\cdots \right)=0,$ (4)

$Q\left(x,t,{u}^{\left(r\right)},{u}^{\left(r+1\right)},\cdots \right)=0,$ (5)

${u}^{\left(r\right)}\left(\xi \right)=v\left(\eta \right)=\frac{p\left(\eta \right)}{q\left(\eta \right)}=\frac{{p}_{m}{\eta }^{m}+{p}_{m-1}{\eta }^{m-1}+\cdots +{p}_{0}}{{q}_{n}{\eta }^{n}+{q}_{n-1}{\eta }^{n-1}+\cdots +{q}_{0}},$ (6)

$\eta =\eta \left(\xi \right)$ 满足一阶微分方程

${\eta }^{\prime }\left(\xi \right)=T=T\left(\xi ,\eta \right),$ (7)

3. 第一类扩展的(3 + 1)维Jimbo-Miwa方程

${u}_{xxxy}+3{u}_{y}{u}_{xx}+3{u}_{x}{u}_{xy}+2{u}_{yt}-3\left({u}_{xz}+{u}_{yz}+{u}_{zz}\right)=0.$ (8)

${a}^{3}b{u}^{\left(4\right)}+6{a}^{2}b{u}^{\prime }{u}^{″}-\left(2b\omega +3ac+3bc+3{c}^{2}\right){u}^{″}=0,$ (9)

${a}^{3}b{u}^{‴}+3{a}^{2}b{\left({u}^{\prime }\right)}^{2}-\left(2b\omega +3ac+3bc+3{c}^{2}\right){u}^{\prime }=0.$ (10)

$r=1$${u}^{\prime }\left(\xi \right)=v\left(\eta \right)$ ，我们得到了方程(8)的变换形式，

${a}^{3}b{T}^{2}{v}^{″}+{a}^{3}bT{T}^{\prime }{v}^{\prime }+3{a}^{2}b{v}^{2}-\left(2b\omega +3ac+3bc+3{c}^{2}\right)v=0.$ (11)

3.1. 当 ${\eta }^{\prime }=\eta$

(11)可以变成

${a}^{3}b{\eta }^{2}{v}^{″}+{a}^{3}b\eta {v}^{\prime }+3{a}^{2}b{v}^{2}-\left(2b\omega +3ac+3bc+3{c}^{2}\right)v=0.$ (12)

$v\left(\eta \right)=\frac{4a{q}_{1}{q}_{2}\eta }{4{q}_{2}^{2}{\eta }^{2}+4{q}_{1}{q}_{2}\eta +{q}_{1}^{2}},\text{\hspace{0.17em}}\omega =\frac{{a}^{3}b-3ac-3bc-3{c}^{2}}{2b}$ (13)

$v\left(\eta \right)=\frac{{p}_{2}\left(4{q}_{2}^{2}{\eta }^{2}-8{q}_{1}{q}_{2}\eta +{q}_{1}^{2}\right)}{{q}_{2}\left(4{\eta }^{2}{q}_{2}^{2}+4{q}_{1}{q}_{2}\eta +{q}_{1}^{2}\right)},$

$a=-\frac{3{p}_{2}}{{q}_{2}},\text{\hspace{0.17em}}\omega =-\frac{3\left(bc{q}_{2}^{3}+{c}^{2}{q}_{2}^{3}-9b{p}_{2}^{3}-3c{p}_{2}{q}_{2}^{2}\right)}{2{q}_{2}^{3}b}.$ (14)

$u\left(x,y,z,t\right)=-\frac{2a{q}_{1}}{2{q}_{2}{\text{e}}^{\xi }+{q}_{1}}+d,$

$\xi =ax+by+cz-\frac{{a}^{3}b-3ac-3bc-3{c}^{2}}{2b}t$

$u\left(x,y,z,t\right)=\frac{6{p}_{2}{q}_{1}}{{q}_{2}\left(2{q}_{2}{\text{e}}^{\xi }+{q}_{1}\right)}+\frac{{p}_{2}}{{q}_{2}}\xi +d,$

$\xi =-\frac{3{p}_{2}}{{q}_{2}}x+by+cz+\frac{3}{2}\frac{\left(bc{q}_{2}^{3}+{c}^{2}{q}_{2}^{3}-9b{p}_{2}^{3}-3c{p}_{2}{q}_{2}^{2}\right)t}{{q}_{2}{}^{3}b},$

d为积分常数，其余的参数均为任意常数。

3.2. 当 ${\eta }^{\prime }=\frac{1}{2}-\frac{1}{2}{\eta }^{2}$

(11)可以变成

$\begin{array}{l}\left(\frac{1}{4}{a}^{3}b{\eta }^{4}-\frac{1}{2}{a}^{3}b{\eta }^{2}+\frac{1}{4}{a}^{3}b\right){v}^{″}+\left(\frac{1}{2}{a}^{3}b{\eta }^{3}-\frac{1}{2}{a}^{3}b\eta \right){v}^{\prime }+3{a}^{2}b{v}^{2}\\ -\left(2b\omega +3ac+3bc+3{c}^{2}\right)v=0.\end{array}$ (15)

$v\left(\eta \right)=-\frac{\left({\eta }^{2}-1\right){p}_{1}}{{q}_{1}},\text{\hspace{0.17em}}\omega =-\frac{3bc{q}_{1}^{3}+3{c}^{2}{q}_{1}^{3}-8b{p}_{1}^{3}+6c{p}_{1}{q}_{1}^{2}}{2{q}_{1}^{3}b}$ (16)

$v\left(\eta \right)=-\frac{\left(3{\eta }^{2}-1\right){p}_{1}}{{q}_{1}},\text{\hspace{0.17em}}\omega =-\frac{3\left(bc{q}_{1}^{3}+{c}^{2}{q}_{1}^{3}+72b{p}_{1}^{3}+6c{p}_{1}{q}_{1}^{2}\right)}{2{q}_{1}^{3}b}.$ (17)

$u\left(x,y,z,t\right)=\frac{2{p}_{1}}{{q}_{1}}\mathrm{tanh}\left(\xi \right)+\frac{2i{p}_{1}}{{q}_{1}\mathrm{cosh}\left(\xi \right)}+d,$

$u\left(x,y,z,t\right)=\frac{2{p}_{1}}{{q}_{1}}\mathrm{coth}\left(\xi \right)+\frac{2{p}_{1}}{{q}_{1}\mathrm{sinh}\left(\xi \right)}+d,$

$\xi =\frac{2{p}_{1}x}{{q}_{1}}+by+cz+\frac{\left(3bc{q}_{1}^{3}+3{c}^{2}{q}_{1}^{3}-8b{p}_{1}^{3}+6c{p}_{1}{q}_{1}^{2}\right)t}{2{q}_{1}^{3}b},$

$u\left(x,y,z,t\right)=\frac{6{p}_{1}}{{q}_{1}}\mathrm{tanh}\left(\xi \right)+\frac{6i{p}_{1}}{{q}_{1}\mathrm{cosh}\left(\xi \right)}-\frac{2{p}_{1}\xi }{{q}_{1}}+d,$

$u\left(x,y,z,t\right)=\frac{6{p}_{1}}{{q}_{1}\mathrm{tanh}\left(\frac{1}{2}\xi \right)}+\frac{2{p}_{1}}{{q}_{1}}\mathrm{ln}\left(\frac{\mathrm{tanh}\left(\frac{1}{2}\xi \right)-1}{\mathrm{tanh}\left(\frac{1}{2}\xi \right)+1}\right)+d,$

$\xi =\frac{6{p}_{1}x}{{q}_{1}}+by+cz+\frac{3\left(bc{q}_{1}^{3}+{c}^{2}{q}_{1}^{3}+72b{p}_{1}^{3}+6c{p}_{1}{q}_{1}^{2}\right)t}{2{q}_{1}^{3}b},$

d为积分常数，其余的参数均为任意常数。

3.3. 当 ${\eta }^{\prime }=1+{\eta }^{2}$

(11)可以变成

$\begin{array}{l}\left({a}^{3}b+2{a}^{3}b{\eta }^{2}+{a}^{3}b{\eta }^{4}\right){v}^{″}+\left(2{a}^{3}b{\eta }^{3}+2{a}^{3}b\eta \right){v}^{\prime }+3{a}^{2}b{v}^{2}\\ -\left(3ac+3bc+2b\omega +3{c}^{2}\right)v=0.\end{array}$ (18)

$v\left(\eta \right)=\frac{\left({\eta }^{2}+1\right){p}_{1}}{{q}_{1}},\text{\hspace{0.17em}}\omega =-\frac{6bc{q}_{1}^{3}+6{c}^{2}{q}_{1}^{3}-b{p}_{1}^{3}-3c{p}_{1}{q}_{1}^{2}}{4{q}_{1}^{3}b}$ (19)

$v\left(\eta \right)=\frac{\left(3{\eta }^{2}+1\right){p}_{1}}{{q}_{1}},\text{\hspace{0.17em}}\omega =-\frac{3\left(2bc{q}_{1}^{3}+2{c}^{2}{q}_{1}^{3}+9b{p}_{1}^{3}-3c{p}_{1}{q}_{1}^{2}\right)}{4{q}_{1}^{3}b}.$ (20)

$u\left(x,y,z,t\right)=\frac{{p}_{1}}{{q}_{1}}\mathrm{tan}\left(\xi \right)+d,$

$\xi =-\frac{{p}_{1}x}{2{q}_{1}}+by+cz+\frac{\left(6bc{q}_{1}^{3}+6{c}^{2}{q}_{1}^{3}-b{p}_{1}^{3}-3c{p}_{1}{q}_{1}^{2}\right)t}{4{q}_{1}^{3}b}$

$u\left(x,y,z,t\right)=\frac{{p}_{1}\left(-2\xi \mathrm{cos}\left(\xi \right)+3\mathrm{sin}\left(\xi \right)\right)}{{q}_{1}\mathrm{cos}\left(\xi \right)}+d,$

$\xi =-\frac{3{p}_{1}x}{2{q}_{1}}+by+cz+\frac{3\left(2bc{q}_{1}^{3}+2{c}^{2}{q}_{1}^{3}+9b{p}_{1}^{3}-3c{p}_{1}{q}_{1}^{2}\right)t}{4{q}_{1}^{3}b},$

d为积分常数，其余的参数均为任意常数。

3.4. 当 ${\eta }^{\prime }=R{\eta }^{2},R\ne 0$

(11)可以变成

${R}^{2}{a}^{3}b{\eta }^{4}{v}^{″}+2{R}^{2}{a}^{3}b{\eta }^{3}{v}^{\prime }+3{a}^{2}b{v}^{2}-\left(3ac+3bc+2b\omega +3{c}^{2}\right)v=0.$ (21)

$v\left(\eta \right)=-2{R}^{2}a{\eta }^{2},\text{\hspace{0.17em}}\omega =-\frac{3c\left(a+b+c\right)}{2b}$ (22)

$v\left(\eta \right)=-\frac{2{R}^{2}a{q}_{0}{\eta }^{2}-{p}_{0}}{{q}_{0}},\text{\hspace{0.17em}}\omega =\frac{3\left(2{a}^{2}b{p}_{0}-ac{q}_{0}-bc{q}_{0}-{c}^{2}{q}_{0}\right)}{2{q}_{0}b}.$ (23)

$u=\frac{2aR}{R\xi +{c}_{0}}+d,\text{\hspace{0.17em}}\xi =ax+by+cz+\frac{3c\left(a+b+c\right)t}{2b}$

$\begin{array}{l}u\left(x,y,z,t\right)=\frac{R{p}_{0}{\xi }^{2}+2Ra{q}_{0}+{c}_{0}{p}_{0}\xi }{{q}_{0}\left(R\xi +{c}_{0}\right)}+d,\\ \xi =ax+by+cz-\frac{3\left(2{a}^{2}b{p}_{0}-ac{q}_{0}-bc{q}_{0}-{c}^{2}{q}_{0}\right)t}{2b{q}_{0}}.\end{array}$

d为积分常数，其余的参数均为任意常数。

4. 第二类扩展的(3 + 1)维Jimbo-Miwa方程

${u}_{xxxy}+3{u}_{y}{u}_{xx}+3{u}_{x}{u}_{xy}+2\left({u}_{yt}+{u}_{xt}+{u}_{zt}\right)-3{u}_{xz}=0.$ (24)

${a}^{3}b{u}^{\left(4\right)}+6{a}^{2}b{u}^{\prime }{u}^{″}-\left(2a\omega +2b\omega +2c\omega +3ac\right){u}^{″}=0,$ (25)

${a}^{3}b{T}^{2}{v}^{″}+{a}^{3}bT{T}^{\prime }{v}^{\prime }+3{a}^{2}b{v}^{2}-\left(2a\omega +2b\omega +2c\omega +3ac\right)v=0.$ (26)

4.1. 当 ${\eta }^{\prime }=\eta$

(26)可以变成

${a}^{3}b{\eta }^{2}{v}^{″}+{a}^{3}b\eta {v}^{\prime }+3{a}^{2}b{v}^{2}-\left(2a\omega +2b\omega +2c\omega +3ac\right)v=0.$ (27)

$v=\frac{4a{q}_{0}{q}_{1}\eta }{{q}_{1}^{2}{\eta }^{2}+4{q}_{0}{q}_{1}\eta +4{q}_{0}^{2}},\text{\hspace{0.17em}}\omega =\frac{a\left({a}^{2}b-3c\right)}{2\left(a+b+c\right)}$ (28)

$v=-\frac{{p}_{1}\left(4{q}_{2}^{2}{\eta }^{2}-8{q}_{1}{q}_{2}\eta +{q}_{1}^{2}\right)}{2{q}_{1}\left(4{q}_{2}^{2}{\eta }^{2}+4{q}_{1}{q}_{2}\eta +{q}_{1}^{2}\right)},\text{\hspace{0.17em}}a=\frac{3{p}_{1}}{2{q}_{1}},\text{\hspace{0.17em}}\omega =-\frac{9{p}_{1}\left(3b{p}_{1}^{2}+4c{q}_{1}^{2}\right)}{8{q}_{1}^{2}\left(2b{q}_{1}+2c{q}_{1}+3{p}_{1}\right)}.$ (29)

$u=-\frac{4a{q}_{0}}{{q}_{1}{\text{e}}^{\xi }+2{q}_{0}}+d,\xi =ax+by+cz-\frac{a\left({a}^{2}b-3c\right)t}{2\left(a+b+c\right)}$

$u=-\frac{3{p}_{1}}{2{q}_{2}{\text{e}}^{\xi }+{q}_{1}}-\frac{{p}_{1}\xi }{2{q}_{1}}+d,\text{\hspace{0.17em}}\xi =\frac{3{p}_{1}}{2{q}_{1}}x+by+cz+\frac{9{p}_{1}\left(3b{p}_{1}^{2}+4c{q}_{1}^{2}\right)t}{8{q}_{1}^{2}\left(2b{q}_{1}+2c{q}_{1}+3{p}_{1}\right)},$

d为积分常数，其余的参数均为任意常数。

4.2. 当 ${\eta }^{\prime }=\frac{1}{2}-\frac{1}{2}{\eta }^{2}$

(26)可以变成

$\begin{array}{l}\left(\frac{1}{4}{a}^{3}b{\eta }^{4}-\frac{1}{2}{a}^{3}b{\eta }^{2}+\frac{1}{4}{a}^{3}b\right){v}^{″}+\left(\frac{1}{2}{a}^{3}b{\eta }^{3}-\frac{1}{2}{a}^{3}b\eta \right){v}^{\prime }+3{a}^{2}b{v}^{2}\\ -\left(2a\omega +2b\omega +2c\omega +3ac\right)v=0.\end{array}$ (30)

$v=\frac{{p}_{3}\left({\eta }^{2}-1\right)}{{q}_{1}},\text{\hspace{0.17em}}a=-\frac{2{p}_{3}}{{q}_{1}},\text{\hspace{0.17em}}\omega =-\frac{{p}_{3}\left(4b{p}_{3}^{2}-3c{q}_{1}^{2}\right)}{{q}_{1}^{2}\left(b{q}_{1}+c{q}_{1}-2{p}_{3}\right)}$ (31)

$v=\frac{{p}_{2}\left(3{\eta }^{2}-4\right)}{3{q}_{0}},\text{\hspace{0.17em}}a=-\frac{2{p}_{2}}{{q}_{2}},\text{\hspace{0.17em}}\omega =-\frac{{p}_{2}\left(8b{p}_{2}^{2}-3c{q}_{0}^{2}\right)}{{q}_{0}^{2}\left(b{q}_{0}+c{q}_{0}-2{p}_{2}\right)}.$ (32)

d为积分常数，其余的参数均为任意常数。

4.3. 当

d为积分常数，其余的参数均为任意常数。

4.4. 当

d为积分常数，其余的参数均为任意常数。

5. 结论

The Transformed Rational Method for Solving Two Extended (3 + 1)-Dimensional Jimbo-Miwa Equations[J]. 应用数学进展, 2018, 07(10): 1247-1255. https://doi.org/10.12677/AAM.2018.710145

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