D. Pelinovsky and G. Schneider

Moving gap solitons in periodic potentials

We address the 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 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.

Gross-Pitaevskii equation, coupled-mode equations, gap solitons in periodic potentials, spatial dynamics, Hamiltonian systems, center and stable manifolds, Implicit Function Theorem