New Exact Travelling Wave Solutions for Fractional Differential Equations in a Shallow Water Waves

In this article, the modified simple equation method is proposed to solve nonlinear space-time fractional differential equations. This method is applied to solve space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation, the space-time fractional generalized reaction duffing model and the space-time fractional potential Kadomtsev-Petviashvili (pKP) equation. The solutions found are hyperbolic and trigonometric function solutions. Some of these solutions are new solutions that are not available in the literature.

Solitary wave solutions, lump solutions and interactional solutions to the (2+1)-dimensional potential Kadomstev–Petviashvili equation

In this paper, the [Formula: see text]-solitary wave solution to the (2[Formula: see text]+[Formula: see text]1)-dimensional potential Kadomstev–Petviashvili (PKP) equation is obtained with the Hirota bilinear method. Via the limit technique of long wave, the [Formula: see text]-lump solution can be derived from resulting [Formula: see text]-solitary wave solution. In addition, interactional solutions consisting of lumps and solitary waves for the PKP equation are obtained, which can describe elastic interactions of lumps and solitary waves. These results are illustrated by graphics of several sample examples.

Symbolic Computation and Construction of Soliton-like Solutions of some Nonlinear Evolution Equations

Based on the computerized symbolic computation system Maple and a Riccati equation, a new generalized Riccati equation expansion method for constructing non-travelling waves and coefficient functions’ soliton-like solutions of nonlinear evolution equations (NEEs) is presented by a general ansatz. Compared with most of the existing tanh methods, namely the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. By use of the method one can not only successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. The cylindrical Korteweg-de Vries (CkdV) equation, a Potential Kadomstev-Petviashvili (PKP) equation, the two-dimensional KdV-Burgers equation are chosen to illustrate our method such that rich new families of exact solutions, including the non-travelling wave soliton-like solutions, singular soliton-like solutions, periodic form solutions are obtained. When taking arbitrary functions of the solutions as some special constants, the known travelling wave solutions of the PKP equation, two-dimensional KdV-Burgers equation can be recovered.