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Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Wednesday, July 14, 19:00 ~ 19:30 UTC-3

Spectral Stability in the nonlinear Dirac equation with Soler-type nonlinearity in dimension $1$.

Hanne Van Den Bosch

Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies $\omega$ in $(0,m)$, where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction. The results that I will discuss concern simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.

Joint work with Danko Aldunate (Pontificia Universidad Católica de Chile), Julien Ricaud (LMU Munich) and Edgardo Stockmeyer (Pontificia Universidad Católica de Chile).

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