A Study on Lump and Interaction Solutions to a (3 + 1)-Dimensional Soliton Equation

Xi-Zhong Liu,Xian-Min Qian,Zhi-Mei Lou,Lamine Thiam

doi:10.1155/2019/9857527

Abstract

Based on bilinear formulation of a (3 + 1)-dimensional soliton equation, lump solution and related interaction solutions are investigated. The lump solutions of the soliton equation are classified into three cases with nonsingularity conditions being given. The interaction solutions between lump and a stripe soliton are obtained in eight cases, which have interesting fusing and fission behaviors with changing time. The interaction solutions of the soliton equation between a lump and a resonant pair of stripe solitons are also given, and we find that the lump just exist for a finite period during the interaction process.

Highlights

In nonlinear systems, soliton solutions are usually exponentially localized in certain directions
Another fascinating feature of lump solutions is that the interaction phase shifts between lump waves are exactly zero [1]
The interactions between lump and various solitons, such as stripe soliton and resonant stripe solitons, have been studied for many integrable nonlinear systems [15,16,17,18,19]. It reveals that most of the interaction solutions between lump and various solitons are completely inelastic [20], while a few of them are elastic [21]. e interaction between lump wave and rogue wave solutions, which have great potential applications in the eld of nonlinear optics [22] and oceanography [23] can be generated by interaction between lump and a pair of resonant kink stripe solitons [8]

Summary

Introduction

In nonlinear systems, soliton solutions are usually exponentially localized in certain directions. Another fascinating feature of lump solutions is that the interaction phase shifts between lump waves are exactly zero [1] In another aspect, the interactions between lump and various solitons, such as stripe soliton and resonant stripe solitons, have been studied for many integrable nonlinear systems [15,16,17,18,19]. E paper is organized as follows: In Section 2, we give general form of lump solutions of the soliton equation (1) in quadratic function form, which can be classified into three cases.

Interaction Solutions between a Lump and a Stripe Soliton

Interaction between a Lump Soliton and a Pair of Stripe Solitons

Conclusion