## Polychromatic Solitary Waves in a Periodic and Nonlinear Maxwell System

- Dmitry Pelinovsky, Gideon Simpson, Michael I. Weinstein
- SIAM J. Applied Dynamical Systems
- 2012

1 Excerpt

- Published 2014

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.

@inproceedings{Pelinovsky2014MovingGS,
title={Moving gap solitons in periodic potentials},
author={Dmitry Pelinovsky and Guido Schneider},
year={2014}
}