Justification of the Coupled Mode Asymptotics for Localized Wavepackets in the Periodic Nonlinear Schr\"odinger Equation

TL;DR

This paper rigorously justifies the use of coupled mode equations to approximate localized wavepackets in the periodic nonlinear Schrödinger equation, supported by error estimates and numerical validation.

Contribution

It provides a rigorous justification and error bounds for the coupled mode approximation of localized wavepackets in the periodic nonlinear Schrödinger equation.

Findings

Coupled mode equations accurately approximate localized wavepackets.

Existence of exponentially localized solitary waves under spectral gap conditions.

Numerical tests confirm the theoretical analysis.

Abstract

We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schroedinger/Gross-Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinear Schroedinger equation. Besides a formal derivation of the CMEs a rigorous justification of the approximation and an error estimate in the supremum norm are provided. Several numerical tests corroborate the analysis.