广义Tanh函数法中Riccati方程和sine-Gordon方程的新解及其新应用

A New Application of the Extended Tanh-function Method and New Solutions of the Riccati Equation and sine-Gordon Equation

非线性偏微分方程的显式解析解,特别是行波解,蕴含了方程的丰富信息,对于描述各种现象的发展规律起着至关重要的作用.本文尝试构造KdV方程多种形式的新显式行波解.首先,利用试探函数法和Matlab计算给出了Riccati方程的许多新显式解析解.其次,运用广义Tanh函数法以及Riccati方程的新解得到了sine-Gordon方程的许多新显式解析解.最后,作为新的应用,把三角函数法结合sine-Gordon方程的新显式解析解并利用简化的变换形式进一步找到了KdV方程的许多新显式行波解.这些结果推广和补充了以往的相关研究成果,特别地,这些方法和新的结果可以用于求解许多非线性偏微分方程的新显式行波解.

The explicit analytical solutions of nonlinear partial differential equations,in particular,the travelling wave solutions,contain rich information about the equations,and they are very important for describing the development of various phenomena.In the paper,many types of new explicit travelling wave solutions are presented for the KdV equation.First,many new explicit analytical solutions of the Riccati equation are given by using the trial function method and Matlab software.Second,many types of new explicit analytical solutions of the sine-Gordon equation are obtained by using the extended tanh-function method and new solutions of Riccati equation.Finally,as a new application,a great many new travelling wave solutions of the KdV equation are provided by using the sine-cosine function method,new solutions of the sine-Gordon equation and its simplified transformation forms.The obtained results extend and complement some relevant existing works.In particular,these methods and obtained new results can be applied to find explicit new travelling wave solutions of many nonlinear partial differential equations.