Exact Solutions to a Generalized Bogoyavlensky‐Konopelchenko Equation via Maple Symbolic Computations

Shou-Ting Chen,Wen-Xiu Ma

doi:10.1155/2019/8787460

Abstract

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky‐Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two‐wave solutions, and polynomial solutions.

Highlights

One of the fundamental problems in the theory of differential equations is to determine a solution of a differential equation satisfying what are known as initial values
The Lie group method and the Hirota bilinear method are among effective approaches for finding exact solutions to nonlinear differential equations
Symbolic computations with Maple are the adopted technique and the Hirota bilinear form is a basis for getting one-wave type and two-wave solutions

Summary

Introduction

One of the fundamental problems in the theory of differential equations is to determine a solution of a differential equation satisfying what are known as initial values. Only the simplest differential equations, often linear, are solvable precisely It is definitely not an easy task for us to find exact solutions to nonlinear differential equations, either ordinary or partial. The Lie group method and the Hirota bilinear method are among effective approaches for finding exact solutions to nonlinear differential equations. Solitons contain various kinds of exact solutions to integrable equations, and taking long wave limits of N-soliton solutions can generate special lumps [16]. Based on its Hirota bilinear form, a few solution ansatze will be analyzed to compute exact solutions to the nonlinear equation and its bilinear counterpart. Starting from the nonlinear equation itself, we will do a thorough symbolic computation by Maple within our capacity to generate a few classes of exact and explicit solutions, including traveling wave solutions, two-wave solutions, and polynomial solutions. Conclusions and remarks will be given in the last section

One-Wave Type and Two-Wave and Polynomial Solutions

Concluding Remarks