Oscillatory instabilities of gap solitons in a repulsive Bose-Einstein condensate

TL;DR

This paper numerically investigates the stability of gap solitons in a one-dimensional Bose-Einstein condensate with a periodic potential, revealing conditions under which they exhibit oscillatory instabilities.

Contribution

It introduces an Evans function approach combined with exterior algebra to detect and analyze weak oscillatory instabilities of gap solitons in the 1D GPE.

Findings

Fundamental gap solitons can become oscillatory unstable at certain chemical potentials.

Number and rate of unstable eigenvalues are characterized.

Stable and unstable complex gap solitons are identified and discussed.

Abstract

The paper is devoted to numerical study of stability of nonlinear localized modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest ("fundamental") gap solitons in the first and in the second spectral gaps can undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The number of unstable eigenvalues and the associated instability rates are described. Several stable and unstable more complex (non-fundamental) gap solitons are also discussed. The results obtained from the Evans function approach are independently confirmed using the direct numerical integration of the GPE.