New Interaction Solutions for a (2 + 1)-Dimensional Vakhnenko Equation

Qing Meng,Bin He

doi:10.1155/2020/5620245

Abstract

In this paper, we focus on the interaction solutions of a (2 + 1)-dimensional Vakhnenko equation. By using Hirota’s transformation combined with the three-wave method and with symbolic computation, some interaction solutions which include interaction solutions between exponential and trigonometric functions and interaction solutions between exponential and trigonometric and hyperbolic functions are presented.

Highlights

It is well known that nonlinear partial differential equations (NPDEs) and their solutions play a significant role in interpreting many important phenomena in nonlinear sciences
A variety of powerful methods are developed for finding the exact solutions of NPDEs, such as Hirota’s method [1, 2], simplified Hirota’s method [3, 4], the Lie symmetry analysis method [5, 6], the simplest equation method [5, 6], the invariant subspace method [7], and the nonlinear steepest descent method [8]
The lump and interaction solutions [9,10,11] have attracted the attention of many scholars because of lump’s applications in nonlinear optics, physics, oceanography, etc, and the interaction solutions are valuable in analyzing the nonlinear dynamics of waves in shallow water and can be used for forecasting the appearance of rogue waves [12, 13]

Summary

Introduction

It is well known that nonlinear partial differential equations (NPDEs) and their solutions play a significant role in interpreting many important phenomena in nonlinear sciences. Vakhnenko and Parkes [17] obtained the two-loop soliton solution for the Vakhnenko equation using Hirota’s bilinear method. Vakhnenko et al [18] derived a Backlund transformation both in the bilinear and in ordinary form for the generalized Vakhnenko equation and found the exact N-soliton solution via the inverse scattering method.

Results

Conclusion