# 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.

**Synopsis:**

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.