Abstract
The paper is devoted to nonlinear localized modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called "codes" of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, the method is applied to compute the profiles of gap solitons for 1D GPE with a cosine potential.
Highlights
The Gross–Pitaevskii equation (GPE) is a commonly recognized model for description of nonlinear matter waves in Bose–Einstein condensates (BECs) [20]
For Region 1 the coding alphabet consists of 3 symbols, and for Region 2 it consists of 5 symbols
More complex gap solitons can be regarded as bound states of several FGS1 taken with the plus or minus sign
Summary
The Gross–Pitaevskii equation (GPE) is a commonly recognized model for description of nonlinear matter waves in Bose–Einstein condensates (BECs) [20]. The nonlinear term σ|Ψ|2Ψ takes into account the interactions between the particles: the case σ = 1 corresponds to the repulsive interparticle interactions, σ = −1 corresponds to the attractive interactions. Both these cases are of physical relevance. The real-valued potential U (x) describes a trap which confines the BEC. In the case of optical confinement of BEC, the potential U (x) is a periodic function
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