Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

Abstract

In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refractive index.

arXiv comment21 pages; minor amendments: more details added to the proof of Proposition 3.3 and to Example 3.4