Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential

TL;DR

This paper rigorously justifies the use of Coupled Mode Equations to approximate gap solitons near a band edge in the 2D Gross-Pitaevskii equation with non-separable periodic potentials, supported by numerical validation.

Contribution

It extends the CME approximation to non-separable potentials in 2D, providing a rigorous mathematical justification and $H^s$ estimates, unlike previous separable potential cases.

Findings

CME approximation valid for non-separable potentials near band edges

Rigorous $H^s$ estimates established for the approximation

Numerical examples confirm new families of gap solitons

Abstract

Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs). Here we study this approximation for the case of the 2D Periodic Nonlinear Schr\"{o}dinger / Gross-Pitaevskii Equation with a non-separable potential of finite contrast. We show that unlike in the case of separable potentials [T. Dohnal, D. Pelinovsky, and G. Schneider, J. Nonlin. Sci. {\bf 19}, 95--131 (2009)] the CME derivation has to be carried out in Bloch rather than physical coordinates. Using the Lyapunov-Schmidt reduction we then give a rigorous justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and even potentials and provide $H^s$ estimates for this approximation. The results are confirmed by numerical examples including some new families of CMEs and gap solitons absent for separable potentials.