D. Pelinovsky and G. Schneider
Moving gap solitons in periodic potentials
Abstract:
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.
Keywords:
Gross-Pitaevskii equation, coupled-mode equations, gap solitons in periodic potentials,
spatial dynamics, Hamiltonian systems, center and stable manifolds, Implicit Function Theorem