Evolution Equations and Control Theory (EECT)

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

Pages: 81 - 100, Volume 2, Issue 1, March 2013      doi:10.3934/eect.2013.2.81

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François Genoud - Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland (email)

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.

Keywords:  Asymptotically linear NLS, global bifurcation, orbital stability, self-focusing waveguide, saturable refractive index.
Mathematics Subject Classification:  Primary: 35Q55; Secondary: 35B32, 35B35.

Received: September 2012;      Revised: December 2012;      Available Online: January 2013.