Localized waves and mixed interaction solutions with dynamical analysis to the Gross–Pitaevskii equation in the Bose–Einstein condensate

Haotian Wang,Wenjun Liu,Qin Zhou,Anjan Biswas

doi:10.1007/s11071-021-06851-z

Abstract

We report a kind of breather, rogue wave, and mixed interaction structures on a variational background height in the Gross–Pitaevskii equation in the Bose–Einstein condensate by the generalized Darboux transformation method, and the effects of related parameters on rogue wave structures are discussed. Numerical simulation can discuss the dynamics and stability of these solutions. We numerically confirm that these are correct and can be reproduced from a deterministic initial profile. Results show that rogue waves and mixed interaction solutions can evolve with a small amplitude perturbation under the initial profile conditions, but breathers cannot. Therefore, these can be used to anticipate the feasibility of their experimental observation.

Highlights

As a new state of matter and a field in physics, the surge of interest in Bose-Einstein condensate (BEC) has been prompted in recent years by researchers [1,2,3,4,5,6,7]
The mean-field Gross-Pitaevskii equation (GPE) is used to describe and understand some nonlinear phenomena, interesting and important properties and characteristics of vortex states [8,9,10,11,12] and localized waves including soliton [13,14,15,16], rogue wave (RW) [17,18,19], breather [20, 21] in BEC, which is one of the main theoretical research methods, and some predictions have been exposed to agree with relevant vortex and localized wave experiments [22,23,24]
We give a set of dynamics of analytical solutions of the GPE in some different potentials with the time-varying nonlinear coefficient

Summary

Introduction

Localized waves are very important nonlinear phenomena, and include, mainly, solitons, RWs and breathers. The exact localized wave solutions and time evolution of them in BECs are described by the GPE with abundant different external potentials. Where the α and β are real parameters in external potentials, and the coefficient u is the nonlinearity strength in the GPE (2). The Eq (2) can describe macroscopic waves in some atom BECs such as Li. for free real nonlinearity strength u. The conclusion in the last section will summarize the whole paper

Darboux transformation

Exact solutions

Discussion

Conclusions