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

TL;DR

This paper proves the existence and orbital stability of standing waves in a one-dimensional asymptotically linear nonlinear Schrödinger equation, providing the first rigorous stability result for this class.

Contribution

It establishes the first rigorous proof of orbital stability for standing waves in the asymptotically linear NLS, along with a global smooth curve of solutions.

Findings

Existence of a global smooth curve of standing waves

Orbital stability of these standing waves

Application to self-focusing waveguides with saturable refractive index

Abstract

In this article we study the one-dimensional, asymptotically linear, non-linear Schr\"odinger 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.