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Center for Nonlinear Studies |
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 Schroedinger 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 and Carleman-type inequalities. The method is applicable to other systems, such as the Dirac-Maxwell system. Some of the results are obtained in collaboration with Nabile Boussaid, University of Franche-Comte, and Stephen Gustafson, University of British Columbia.
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