Traveling Solitary Waves in the Periodic Nonlinear Schrödinger Equation with Finite Band Potentials
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Date
2013-05-16
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Abstract
The paper studies asymptotics of moving gap solitons in nonlinear periodic structures
of finite contrast ("deep grating") within the one dimensional periodic nonlinear Schr¨odinger equation
(PNLS). Periodic structures described by a finite band potential feature transversal crossings of band
functions in the linear band structure and a periodic perturbation of the potential yields new small
gaps. An approximation of gap solitons in such a gap is given by slowly varying envelopes which
satisfy a system of generalized Coupled Mode Equations (gCME) and by Bloch waves at the crossing
point. The eigenspace at the crossing point is two dimensional and it is necessary to select Bloch
waves belonging to the two band functions. This is achieved by an optimization algorithm. Traveling
solitary wave solutions of the gCME then result in nearly solitary wave solutions of PNLS moving at
an O(1) velocity across the periodic structure. A number of numerical tests are performed to confirm
the asymptotics.
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coupled mode equations, envelope approximation, finite band potential, Gross-Pitaevskii equation, Lamé's equation, moving gap soliton, nonlinear Schrödinger equation, periodic structure with finite contrast