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

Abstract

We address the Gross–Pitaevskii equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works, we show that the intersite cubic nonlinear terms in the discrete nonlinear Schrödinger (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 have the quintic nonlinear term, which correctly describes bifurcation of gap solitons in the semi-infinite gap.

Keywords:

• Gross–Pitaevskii equation
• discrete nonlinear Schrödinger equation
• gap solitons
• bifurcations
• semi-classical limit
• Wannier functions

AMS Subject Classifications::

• 35Q55
• 37K60

Acknowledgements

J. Belmonte-Beitia is partially supported by grants PCI08-0093 (Consejería de Educación y Ciencia de la Junta de Comunidades de Castilla-La Mancha, Spain), PRINCET and FIS2006-04190 (Ministerio de Educación y Ciencia, Spain). J. Belmonte-Beitia would also like to thank the Department of Mathematics at McMaster University for hospitality during his visit. D. Pelinovsky is partially supported by the NSERC grant.