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

Abstract:

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.