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Nondegeneracy and stability of periodic traveling waves in a fractional NLS equation

Banff International Research Station for Mathematical Innovation and Discovery

In the stability and blowup for traveling or standing waves in nonlinear Hamiltonian dispersive equations, the non-degeneracy of the linearization about such a wave is of paramount importance. That is, one must verify the kernel of the second variation of the Hamiltonian is generated by the continuous symmetries of the PDE. The proof of this property can be far from trivial, especially in cases where the dispersion admits a nonlocal description where shooting arguments, Sturm-Liouville theories, and other ODE methods may not be applicable. In this talk, we discuss the non degeneracy and nonlinear orbital stability of antiperiodic traveling wave solutions to a class of defocusing NLS equations with fractional dispersion. Key to our analysis is the development of a ground state theory and oscillation theory for linear periodic, fractional Schrodinger operators with antiperiodic boundary conditions. This is joint work with Kyle Claassen (KU).

Mathematics; Partial differential equations; Dynamical systems and ergodic theory; Functional analysis

video/mp4

Author affiliation: University of Kansas

2017-12-18

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64051

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/