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

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.

arXiv comment12 pages, 7 figures; accepted for Physica D

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• Bright and dark solitons in the systems with strong light-matter coupling: exact solutions and numerical simulations
• Adiabatic theory of one-dimensional curved polariton waveguides
• Formation of nonlinear modes in one-dimensional quasiperiodic lattices with a mobility edge
• Stability of gap soliton complexes in the nonlinear Schrödinger equation with periodic potential and repulsive nonlinearity