Nonlinear stability of stationary periodic solutions of the focusing NLS equation

The spectral instabilities of the stationary periodic solutions of the focusing NLS equation were 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 dierent Krein signature calculations. Similar, more recent results for the sine-Gordon equation will be shown as well.Non UBCUnreviewedAuthor affiliation: University of WashingtonFacult