Localized modes in the Gross-Pitaevskii equation with a parabolic trapping potential and a nonlinear lattice pseudopotential

TL;DR

This paper investigates localized modes in a one-dimensional Gross-Pitaevskii equation with a harmonic trap and a nonlinear lattice pseudopotential, revealing new stable modes and stabilization effects relevant to Bose-Einstein condensates.

Contribution

It introduces the analysis of localized modes in a model with a nonlinear lattice pseudopotential, showing creation of new modes and stabilization of unstable ones.

Findings

New localized mode families created by the lattice pseudopotential

Stabilization of previously unstable localized modes

Unstable modes evolve into pulsating trapped modes

Abstract

We study localized modes (LMs) of the one-dimensional Gross-Pitaevskii/nonlinear Schr\"{o}dinger equation with a harmonic-oscillator (parabolic) confining potential, and a periodically modulated coefficient in front of the cubic term (nonlinear lattice pseudopotential). The equation applies to a cigar-shaped Bose-Einstein condensate loaded in the combination of a magnetic trap and an optical lattice which induces the periodic pseudopotential via the Feshbach resonance. Families of stable LMs in the model feature specific properties which result from the interplay between spatial scales introduced by the parabolic trap and the period of the nonlinear pseudopotential. Asymptotic results on the shapes and stability of LMs are obtained for small-amplitude solutions and in the limit of a rapidly oscillating nonlinear pseudopotential. We show that the presence of the lattice pseudopotential may result in: (i) creation of new LM families which have no counterparts in the case of the uniform nonlinearity; (ii) stabilization of some previously unstable LM species; (iii) evolution of unstable LMs into a pulsating mode trapped in one well of the lattice pseudopotential.