Orbitally stable standing waves for the asymptotically linearone-dimensional NLS

François Genoud

doi:10.3934/eect.2013.2.81

Article Contents

2013, Volume 2, Issue 1: 81-100. Doi: 10.3934/eect.2013.2.81

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

François Genoud^1,

1.
Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS

Received: September 2012

Revised: December 2012

Published: March 2013

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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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B32, 35B35.

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