Bifurcation and orbital stability of standing waves for some nonlinear Schr\"odinger equations

Bifurcation and orbital stability of standing waves for some nonlinear Schr\"odinger equations

Wed, 11/03/2009
13:00 		François Genoud (OxPDE, University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Bifurcation and orbital stability of standing waves for some nonlinear Schrödinger equations
The aim of my talk is to present the work of my PhD Thesis and my current research. It is concerned with local/global bifurcation of standing wave solutions to some nonlinear Schrödinger equations in $ \mathbb{R}^N \ (N\geq1) $ and with stability properties of these solutions. The equations considered have a nonlinearity of the form $ V(x)|\psi|^{p-1}\psi $, where $ V:\mathbb{R}^N\to\mathbb{R} $ decays at infinity and is subject to various assumptions. In particular, $ V $ could be singular at the origin. Local/global smooth branches of solutions are obtained for the stationary equation by combining variational techniques and the implicit function theorem. The orbital stability of the corresponding standing waves is studied by means of the abstract theory of Grillakis, Shatah and Strauss.