Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: coding and method for computation

TL;DR

This paper introduces a numerical method to reconstruct localized gap soliton modes in the 1D Gross-Pitaevskii equation with periodic potential using symbolic coding, facilitating the analysis of these nonlinear localized states.

Contribution

It presents a new computational technique that reconstructs gap soliton profiles from symbolic codes, advancing the analysis of localized modes in nonlinear Schrödinger equations.

Findings

Successfully applied the method to cosine potential

Validated the coding-based reconstruction of gap solitons

Enhanced understanding of localized mode structures

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.