Moving gap solitons in periodic potentials
Abstract
We address existence of moving gap solitons (traveling localized solutions) in the Gross-Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit localized solutions of the coupled-mode system. We show however that exponentially decaying traveling solutions of the Gross-Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross--Pitaevskii equation with a single bump surrounded by oscillatory tails on a finite large interval of the spatial scale is proven by using these technique. We also show generality of oscillatory tails in other nonlinear equations with a periodic potential.
Keywords
Cite
@article{arxiv.0704.2123,
title = {Moving gap solitons in periodic potentials},
author = {Dmitry Pelinovsky and Guido Schneider},
journal= {arXiv preprint arXiv:0704.2123},
year = {2009}
}