Bifurcation of gap solitons in periodic potentials with a sign-varying nonlinearity coefficient

TL;DR

This paper investigates the bifurcation of gap solitons in the Gross--Pitaevskii equation with a periodic potential and sign-changing nonlinearity, revealing that previous assumptions about intersite nonlinear terms are invalid and that quintic nonlinearities are more accurate.

Contribution

It corrects previous misconceptions by showing that intersite cubic nonlinear terms are not applicable and identifies quintic nonlinearities as the proper description for bifurcations.

Findings

Intersite cubic nonlinear terms are invalid beyond tight-binding approximation.

Quintic nonlinear terms accurately describe bifurcation of gap solitons.

Previous claims about intersite cubic terms are contradicted by this analysis.

Abstract

We address the Gross--Pitaevskii (GP) equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works of Abdullaev {\em et al.} [PRE {\bf 77}, 016604 (2008)] and Smerzi & Trombettoni [PRA {\bf 68}, 023613 (2003)], we show that the intersite cubic nonlinear terms in the discrete nonlinear Schr\"odinger (DNLS) equation appear beyond the applicability of assumptions of the tight-binding approximation. Instead of these terms, for an even linear potential and an odd nonlinearity coefficient, the DNLS equation and other reduced equations for the semi-infinite gap have the quintic nonlinear term, which correctly describes bifurcation of gap solitons.