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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 1207 - 1229, Volume 22, Issue 4, June 2017      doi:10.3934/dcdsb.2017059

 
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Georgy L. Alfimov - Moscow Institute of Electronic Engineering, Zelenograd, Moscow, 124498, Russian Federation (email)
Pavel P. Kizin - Moscow Institute of Electronic Engineering, Zelenograd, Moscow, 124498, Russian Federation (email)
Dmitry A. Zezyulin - Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Ed. C8, Lisboa P-1749-016, Portugal (email)

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.

Keywords:  Gross-Pitaevskii equation, gap solitons, nonlinear modes, coding.
Mathematics Subject Classification:  35Q55, 35C08, 34C41, 37B10, 65P99.

Received: May 2016;      Revised: September 2016;      Available Online: February 2017.

 References