The Exact Traveling Wave Solution of the Integral Differential KP Hierarchy Equation

Jia-qian SONG; Xiao-hua LIU

doi:10.13718/j.cnki.xsxb.2019.10.005

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By means of the integral transformation and traveling wave transformation, the KP hierarchical equation can be turned into the equivalent ordinary differential equation. With the help of the extended trial equation method, the problem of solving ordinary differential equation is transformed into that of solving algebraic equation system. According to the solutions of algebraic equation system and the extended trial equation, we have obtained the exact expressions of bell-shaped solution, triangular solution, hyperbolic solution and elliptic function solution of the KP hierarchical equation. The explicit expressions of these solutions are obtained for the first time. With the extended trial equation method, many new exact solutions of nonlinear partial differential equations can be obtained effectively.

Corresponding author: Xiao-hua LIU

Received Date: 14/12/2018

Available Online: 20/10/2019

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Abstract: By means of the integral transformation and traveling wave transformation, the KP hierarchical equation can be turned into the equivalent ordinary differential equation. With the help of the extended trial equation method, the problem of solving ordinary differential equation is transformed into that of solving algebraic equation system. According to the solutions of algebraic equation system and the extended trial equation, we have obtained the exact expressions of bell-shaped solution, triangular solution, hyperbolic solution and elliptic function solution of the KP hierarchical equation. The explicit expressions of these solutions are obtained for the first time. With the extended trial equation method, many new exact solutions of nonlinear partial differential equations can be obtained effectively.

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非线性偏微分方程在数学、物理、化学以及其它学科和工程领域起着非常重要的作用.很多学者研究了非线性偏微分方程，因此许多求解非线性偏微分方程的方法也发展起来.比如Jacobi椭圆函数展开法^[1]、辅助函数法^[2]、指数函数法^[3]、sine-cosine法^[4]、G′/G展开法^[5-6]、Bäcklund变换法^[7]等.扩展试验方程法是一种常用于求解非线性偏微分方程精确解的方法，文献[8]利用扩展试验方程法得到了广义Benjamin方程和Burger-Kdv方程的精确行波解，文献[9]通过扩展试验方程法考虑了非线性耦合Schrodinger Boussinesq偏微分方程的精确行波解.

本文主要探讨积分微分KP层次方程

的精确行波解，其中

KP层次方程在数学物理和工程中有许多重要的应用，是一类非常重要的非线性偏微分方程.文献[10]利用重复齐次平衡法得到了(3+1)维KP方程的孤子解和周期解，文献[11]通过G′/G-展开法得到了(2+1)维KP方程的行波解.文献[12]通过扩展tanh法得到了耦合KP方程的行波解.我们将对方程(1)进行积分变换，之后再对其进行行波变换，然后利用扩展试验方程法对方程(1)的精确行波解进行研究.

1. 方程(1)的行波变换

在方程(1)中做变换u=v_xx，可将积分微分KP层次方程(1)转换成如下的方程

对方程(2)进行行波变换，再对其进行一次积分，令v(x，y，t)=v(ξ)，其中ξ=x+y－kt，k是波速，方程(2)可化为

其中$v_{\xi\xi}=\frac{\partial^{2} v}{\partial \xi^{2}}, v^{(4)}=\frac{\partial^{4} v}{\partial \xi^{4}}$，C是积分常数.令w=v_ξξ，方程(3)可化为如下等价的方程

综上推导可知，方程(4)的解为w=v_ξξ，而方程(1)的解为u=v_xx，显然方程(1)的解与方程(4)的解具有相同形式.为此下面只需对方程(4)进行研究.

3. 总结

利用扩展试验方程法得到了积分微分KP层次方程的钟状解、三角函数解、双曲函数解和椭圆函数解的精确表达式.通过查阅文献，这些解的显示表达式是首次求出的，可以看出扩展试验方程法是求解非线性偏微分方程的有效方法，可以用于求解数学物理中的非线性偏微分方程.

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The Exact Traveling Wave Solution of the Integral Differential KP Hierarchy Equation

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