Spectral stability for periodic waves in some Hamiltonian systems

PDE Seminar
Tuesday, September 12, 2023 - 3:30pm for 1 hour (actually 50 minutes)
Skiles 005
Atanas Stefanov – University of Alabama at Birmingham – stefanov@uab.eduhttps://www.uab.edu/cas/mathematics/people/faculty-directory/atanas-stefanov
Gong Chen

A lot of recent work in the theory of partial differential equations has focused on the existence and stability properties of special solutions for Hamiltonian PDE’s.  

We review some recent works (joint with Hakkaev and Stanislavova), for spatially periodic traveling waves and their stability properties. We concentrate on three examples, namely the Benney system, the Zakharov system and the KdV-NLS model. We consider several standard explicit solutions, given in terms of Jacobi elliptic functions. We provide explicit and complete description of their stability properties. Our analysis is based on the careful examination of the spectral properties of the linearized operators, combined with recent advances in the Hamiltonian instability index formalism.