Seminar 30.11.2017: Bernard Deconinck

Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar

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Seminar 30.11.2017: Bernard Deconinck

Postby quittner » Thu Nov 23, 2017 11:17 am

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 30.11.2017 at 14:00 Lecture room M-223

Bernard Deconinck (University of Washington):
The stability of solutions of integrable equations

Examining the stability of solutions of nonlinear PDEs continues to be
an active area of research. Very few instances lend themselves to
explicit results for even spectral and linear stability, let alone
orbital (nonlinear) stability. Using the Lax pair structure of
integrable equations, much progress has been made recently on the
stability or instability of solutions of integrable problems.

After introducing the necessary concepts, I will discuss our recent
work on the stability of standing wave solutions of the focusing NLS
equation. The spectral stability of these solutions was completely
characterized recently. The crux of this characterization was the
analysis of the non-self adjoint Lax pair for the focusing NLS
equation. Although all solutions are unstable in the class of bounded
perturbations, different solutions were found to be spectrally stable
with respect to certain classes of periodic perturbations, with period
an integer multiple of the solution period. We prove that all
solutions that are spectrally stable are also (nonlinearly) orbitally
stable, using different Krein signature calculations. Time permitting, more
recent results for the sine-Gordon equation will be shown as well.
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