Bright and dark solitons in a quasi 1D Bose-Einstein condensates modelled by 1D Gross-Pitaevskii equation with time-dependent parameters

TL;DR

This paper derives exact bright and dark soliton solutions for a 1D Bose-Einstein condensate modeled by a time-dependent Gross-Pitaevskii equation, exploring various trapping potentials and parameter modulations.

Contribution

It transforms the 1D Gross-Pitaevskii equation into an integrable nonlinear Schrödinger form and analyzes soliton dynamics under different trap and parameter modulations.

Findings

Existence of periodic oscillating solitons

Collapse and revival phenomena in condensates

Formation of two-soliton bound states

Abstract

We investigate the exact bright and dark solitary wave solutions of an effective one dimensional (1D) Bose-Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature P\'erez-Garc\'ia et al. [Physica D 221 (2006) 31], Serkin et al. [Phys. Rev. Lett. 98 (2007) 074102], G\"urses [arXiv:0704.2435] and Kundu [Phys. Rev. E 79 (2009) 015601], we point out that the effective 1D equation res ulting from the Gross-Pitaevskii (GP) equation can be transformed into the stand ard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schr\"o dinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton like bound state, as the atomic scattering length and gain/loss term are varied. However when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials.