Bright–dark solitons in the space-shifted nonlocal coupled nonlinear Schrödinger equation

Ping Ren,Jiguang Rao

doi:10.1007/s11071-022-07269-x

Abstract

Multiple bright–dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schrödinger equation are constructed by using the bilinear (Kadomtsev–Petviashvili) KP hierarchy reduction method. It is found that the bright–dark two-soliton only occurs elastic collisions. Upon their amplitudes, the bright two solitons only admit one pattern whose amplitude are equal, and the dark two solitons have three different non-degenerated patterns and two different degenerated patterns. The bright–dark four-soliton is the superposition of the two-soliton pairs and can generate the bound-state solitons. The multiple double-pole bright–dark soliton solutions are derived through a long wave limit of the obtained bright–dark soliton solutions, and their collision dynamics are also investigated.

Highlights

The coupled nonlinear Schrodinger (CNLS) or CNLS-type equations have attracted considerable attentions because of their wide applications in a wide scope of physical fields, spanning from nonlinear optics to, water waves, atomic condensates, plasma physics, and others [1–5]
The first example is the nonlocal NLS equation, which was introduced by Ablowitz and Musslimani from the particular reductions of the Ablowitz– Kaup–Newell–Segur (AKNS) hierarchy [7]
We study the space-shifted nonlocal CNLS equation iut + uxx + 2 δuu∗(x0 − x, t) + γvv∗(x0 − x, t) u = 0, ivt + vxx + 2 δuu∗(x0 − x, t) + γvv∗(x0 − x, t) v = 0, (2)

Summary

Introduction

The coupled nonlinear Schrodinger (CNLS) or CNLS-type equations have attracted considerable attentions because of their wide applications in a wide scope of physical fields, spanning from nonlinear optics to, water waves, atomic condensates, plasma physics, and others [1–5]. The first example is the nonlocal NLS equation, which was introduced by Ablowitz and Musslimani from the particular reductions of the Ablowitz– Kaup–Newell–Segur (AKNS) hierarchy [7] Since their seminal works, hierarchies of nonlocal integrable equations and their exact solutions were proposed and studied [8–21]. – The multiple bright-dark soliton solutions in terms of determinants for the space-shifted nonlocal CNLS equation (2) vua the bilinear KP hierarchy reduction.

Multiple bright-dark solitons in the space-shifted nonlocal CNLS equation

Multiple bright-dark soliton solutions in forms of determinants

Dynamics of the bright-dark soliton interactions

Conclusion