Exact Combined Solutions for the (2+1)-Dimensional Dispersive Long Water-Wave Equations

Yi Wei,Lu-Feng Gu,Zhu-Yan Shao,Xing-Qiu Zhang,Xiao-Feng Yang

doi:10.1155/2020/3707924

Abstract

The homogeneous balance of undetermined coefficient (HBUC) method is presented to obtain not only the linear, bilinear, or homogeneous forms but also the exact traveling wave solutions of nonlinear partial differential equations. Linear equation is obtained by applying the proposed method to the (2+1)-dimensional dispersive long water-wave equations. Accordingly, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the (2+1)-dimensional dispersive long water-wave equations are obtained directly. The HBUC method, which can be used to handle some nonlinear partial differential equations, is a standard, computable, and powerful method.

Highlights

Nonlinear partial differential equations (NLPDEs) are used to describe a variety of phenomena in physics [1, 2], thermodynamics [3], fluid dynamics [4, 5], and practical engineering [6, 7] and in several other fields [8]
How to obtain the traveling wave solutions for NLPDEs is very important in the nonlinear phenomena [1, 9, 10]
There is no unified approach, which can be dealt with all NLPDEs

Summary

Introduction

Nonlinear partial differential equations (NLPDEs) are used to describe a variety of phenomena in physics [1, 2], thermodynamics [3], fluid dynamics [4, 5], and practical engineering [6, 7] and in several other fields [8]. Some exact solutions are omitted by using Hirota’s bilinear method, the threewave method, and the ðG′/GÞ-expansion method if the NLPDEs can be linearized To solve this problem, the HBUC method is proposed to derive the linear forms of NLPDEs. In this paper, the (2 + 1)-dimensional dispersive long water-wave equations (DLWEs) [33, 34] are investigated as follows: uyt. The linear equation of the DLWEs is derived by the HBUC method. The multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the DLWEs are obtained directly.

Description of the HBUC Method

Conclusions