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Mathematics Colloquium

Friday, 15 November, 2013

SPEAKER: Prof. Andrew Comech, Texas A&M University and IITP (Moscow)

TITLE: Linear instability of solitary waves in nonlinear Dirac equation

ABSTRACT: We study the linear instability of solitary wave solutions phi(x)e^{-iomega t} to the nonlinear Dirac equation. That is, we linearize the equation at a solitary wave and examine the presence of eigenvalues with positive real part.

To limit the possibility of bifurcations of eigenvalues from the continuous spectrum, we use the limiting absorption principle and the Carleman-type estimates of Berthier--Georgescu.

We also show that the border of the linear instability region is described not only by the Vakhitov-Kolokolov condition Q'(omega)=0, obtained in the NLS context, but also by the energy vanishing condition, E(omega)=0. Here E, Q are the energy and the charge of a solitary wave with frequency omega.

Some of the results are obtained in collaboration with Nabile Boussaid, Université de Franche-Comté,

Stephen Gustafson, University of British Columbia, Gregory Berkolaiko and Alim Sukhtayev, Texas A&M University.

Preprints: arXiv:1209.1146, arXiv:1211.3336, arXiv:1306.5150

Refreshments available at 3:10 p.m.


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