Coupled mode equations and gap solitons in higher dimensions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Dohnal, Tomas¹  Wahlers, Lisa² 

[1] Martin Luther Univ Halle Wittenberg, Inst Math, D-06099 Halle, Saale, Germany

[2] Tech Univ Dortmund, Fak Math, Vogelpothsweg 87, D-44227 Dortmund, Germany

关键词: Wave-packets;    Coupled mode equations;    Periodic media;    Gap soliton;    Approximation error;    Gross-Pitaevskii;   

DOI  :  10.1016/j.jde.2020.01.037

来源: Elsevier

【 摘 要 】

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 is an element of 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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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