Lump Solutions of a Nonlinear PDE Combining with a New Fourth-Order TermDx2Dt2∗

Wen-Xiu Ma,Liqin Zhang,Yehui Huang

doi:10.1155/2020/3542320

Abstract

A nonlinear PDE combining with a new fourth-order termDx2Dt2is studied. Adding three new fourth-order derivative terms and some second-order derivative terms, we formulate a combined fourth-order nonlinear partial differential equation, which possesses a Hirota’s bilinear form. The class of lump solutions is constructed explicitly through Hirota’s bilinear method. Their dynamical behaviors are analyzed through plots.

Highlights

In the theory of differential equations, one of the fundamental problems is the Cauchy problem to determine a solution of a differential equation satisfying what are known as initial data
The isomonodromic transform method and the inverse scattering transform method have been created for handling Cauchy problems for nonlinear ordinary and partial differential equations, respectively [1, 2]
A soliton solution is an exact solution determined by exponentially localized functions, which localized in all directions both in time and in space

Summary

Introduction

In the theory of differential equations, one of the fundamental problems is the Cauchy problem to determine a solution of a differential equation satisfying what are known as initial data. A lump solution is a kind of exact solutions of partial differential equations, obtained from soliton theory by taking long wave limits [1]. Various studies show the existence of interaction solutions between lumps and other kinds of exact solutions to nonlinear integrable equation [2, 18,19,20,21,22,23]. Adding three new fourth-order derivative terms and some second-order derivative terms, we formulate a combined fourth-order nonlinear partial differential equation, which possesses a Hirota’s bilinear form. Based on a bilinear transformation, lump solutions are obtained through symbolic computation with Maple This new term D2xD2t is the second derivative with respect to time, which really makes the calculation more difficult.

Bilinear Form

Lump Solutions

Conclusion and Remarks