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Coupled Mode Equations and Gap Solitons in Higher Dimensions. (arXiv:1810.04944v1 [math.AP])
来源于:arXiv
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 $N\in \mathbb{N}$ 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 $L^1$-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. 查看全文>>