Abstract
We study wave-packets in nonlinear periodic media in arbitrary (d) spatial dimension, modeled by the cubic Gross-Pitaevskii equation. In the asymptotic setting of small and broad wave-packets with N∈N carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an L1-norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the d-dimensional CMEs, we discuss spectral gaps of the CMEs. For N=4 and d=2 a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for d>1 and for the considered values of N due to the absence of a spectral gap in the standard moving frame variables.
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