Stability and long time dynamics of Hamiltonian PDEs

Research Horizons Seminar
Wednesday, April 22, 2015 - 12:05
1 hour (actually 50 minutes)
Skiles 006
Georgia Tech
Many physical models without dissipation can be written in a Hamiltonian form. For example, nonlinear Schrodinger equation for superfluids and Bose-Einstein condensate, water waves and their model equations (KDV, BBM, KP, Boussinesq  systems...), Euler equations for inviscid fluids, ideal MHD for plasmas in fusion devices, Vlasov models for collisionless plasmas and galaxies, Yang-Mills equation in gauge field theory etc. There exist coherent structures (solitons, steady states, traveling waves, standing waves etc) which play an important role on the long time dynamics of these models. First, I will describe a general framework to study linear stability (instability) when the energy functional is bounded from below. For the models with indefinite energy functional (such as full water waves), approaches to find instability criteria will be mentioned. The implication of linear instability (stability) for nonlinear dynamics will be also briefly discussed.