扩展(G′G)—展开法及其在非线性发展方程(组)中的应用

An Extended(G′ G )—expansion Method and its Applications to the Nonlinear Evolution Equations

以逆Cole-Hopf变换为辅助,从一般Riccati方程的已知解构造一类二阶线性常微分方程的一些新精确解.基于该二阶线性常微分方程及其新精确解,在王的(G′G)—展开法和tanh-coth方法的框架下,推出扩展(G′G)—展开法.为检验方法的直接、简洁和有效性,把它应用到Broer-Kaup方程组,得丰富的新行波解,其中包括双曲函数解、三角函数解、指数函数解和有理函数解.该方法可适用于数学物理中的其它非线性发展方程(组).

With the help of the inverse Cole-Hopf transformation, we obtain some new exact solutions of a second order linear ordinary differential equation from known solutions of the general Riccati equation. Based on this second order linear ordinary differential equation and its new exact solutions, an extended （ G＇/G ）--expansion method is proposed under the framework of Wang＇s （ G＇/G ）-expansion method and the tanh-coth method. Being straightforward, concise and effective, the method is applied to the Broer-Kaup equations. As a result, rich new travelling wave solutions are obtained, which contain the hyperbolic function solutions, the trigonometric function solutions, the exponential function solutions and the rational function solutions. This method can be applied to other nonlinear evolution equations in mathematical physics.