2018/10/11 by Dohnal, Tomas, Wahlers, Lisa
#35L71 #35Q55 #35Q60 #41A60 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS)
paper · doi:10.48550/arxiv.1810.04944
We study waves-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 waves-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.