Resonant collisions among two-dimensional localized waves in the Mel’nikov equation

Yinshen Xu,Dumitru Mihalache,Jingsong He

doi:10.1007/s11071-021-06880-8

Abstract

We study the resonant collisions among different types of localized solitary waves in the Mel’nikov equation, which are described by exact solutions constructed using Hirota direct method. The elastic collisions among different solitary waves can be transformed into resonant collisions when the phase shifts of these solitary waves tend to infinity. First, we study the resonant collision among a breather and a dark line soliton. We obtain two collision scenarios: (i) the breather is semi-localized in space and is not localized in time when it obliquely intersects with the dark line soliton, and (ii) the breather is semi-localized in time and is not localized in space when it parallelly intersects with the dark line soliton. The resonant collision of a lump and a dark line soliton, as the limit case of resonant collision of a breather and a dark line soliton, shows the fusing process of the lump into the dark line soliton. Then, we investigate the resonant collision among a breather and two dark line solitons. In this evolution process, we also obtain two dynamical behaviors: (iii) when the breather and the two dark line solitons obliquely intersect each other, we get that the breather is completely localized in space and is not localized in time, and (iv) when the breather and the two dark line solitons are parallel to each other, we get that the breather is completely localized in time and is not localized in space. The resonant collision of a lump and two dark line solitons is obtained as the limit case of the resonant collision among a breather and two dark line solitons. In this special case, the lump first detaches from a dark line soliton and then disappears into the other dark line soliton. Eventually, we also investigate the intriguing phenomenon that when a resonant collision among a breather and four dark line solitons occurs, we get the interesting situation that two of the four dark line solitons are degenerate and the corresponding solution displays the same shape as that of the resonant collision among a breather and two dark line solitons, except for the phase shifts of the solitons, which are not only dependent of the parameters controlling the waveforms of the solitons and the breather, but also dependent of some parameters irrelevant to the waveforms.

Highlights

The theoretical and experimental studies of nonlinear dynamics and unique interaction scenarios of various types of nonlinear waveforms are flourishing due to diverse applications in physics and engineering
We study the resonant collisions among different types of localized solitary waves in the Mel’nikov equation, which are described by exact solutions constructed using Hirota’s direct method
We have found that the elastic collisions among different waveforms can be transformed into resonant collisions when the phase shifts of the involved solitary waves go to infinity

Summary

Introduction

The theoretical and experimental studies of nonlinear dynamics and unique interaction scenarios of various types of nonlinear waveforms are flourishing due to diverse applications in physics and engineering. We point out that families of exact solutions and the interaction scenarious between the corresponding waveforms have been recently investigated for other nonlinear partial differential equations that are relevant in several physics and engineering settings; see, for example, Refs. The Hirota direct method is used in order to obtain the exact solutions of the Mel’nikov equation (1) that describe the resonant collision scenarios between breathers and one dark line solitons and between lumps and one dark line solitons. In Section (2), we focus on deriving the exact solution describing the resonant collision between the breather and one dark line soliton and between the lump and one dark line soliton. We study the resonant collision among a breather and a dark line soliton and the resonant collision of a lump and a dark line soliton based on the solutions (82) given in Appendix. We point out that the latter resonant collision is a limit case of the former one

The resonant collision among a breather and a dark line soliton

The resonant collision among a lump and a dark line soliton

Resonant collision of a breather and two dark line solitons

Resonant collision of a lump and two dark line solitons

Resonant collision of a breather and four dark line solitons

Summary and discussion