Generalized Riccati equation expansion method and its application to the (3 + 1)-dimensional Jumbo–Miwa equation

Abstract

Based on the computerized symbolic system Maple and a Riccati equation, a generalized Riccati equation expansion method for constructing soliton-like solutions of non-linear evolution equations (NEEs) is presented by a new general ansätz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Making use of the method, we study the the (3 + 1)-dimensional Jumbo–Miwa equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.