Analysis Seminar

Linear Instability of Solitary Waves in Nonlinear Dirac Equation

Speaker: Andrew Comech, Texas A&M University

Location: Warren Weaver Hall 1302

Date: Thursday, September 27, 2012, 11 a.m.


We study the linear instability of solitary wave solutions to the nonlinear Dirac equation (known to physicists as the Soler model). That is, we linearize the equation at a solitary wave and examine the presence of eigenvalues with positive real part.

We show that the linear instability of the small amplitude solitary waves is described by the Vakhitov-Kolokolov stability criterion which was obtained in the context of the nonlinear Schrödinger equation: small solitary waves are linearly unstable in dimensions 3, and generically linearly stable in 1D.

A particular question is on the possibility of bifurcations of eigenvalues from the continuous spectrum; we address it using the limiting absorption principle and the Hardy-type estimates.

The method is applicable to other systems, such as the Dirac-Maxwell system.

Some of the results are obtained in collaboration with Nabile Boussaid, Université de Franche-Comté, and Stephen Gustafson, University of British Columbia.