Special topics course offered in Fall 2018 by Zhiwu Lin on "Dynamics of Hamiltonian PDEs."
A lot of conservative PDEs can be written in a Hamiltonian form. They include lots of dispersive PDEs such as nonlinear Schrodinger equation, Gross-Pitaveski equation, gravity water waves and various long wave models (KDV, BBM, KP equations etc). Hamiltonian structures are also found in many models from fluids and plasmas including Euler equations for incompressible ideal fluids, Vlasov models for collisionless plasmas, Euler-Einstein equation for neutron stars etc., usually with degenerate symplectic structures. We will discuss dynamical properties of these Hamiltonian PDEs. Topics to be discussed include: linear stability/instability criterion of coherent states (steady states, traveling waves, standing waves etc.); methods to prove nonlinear stability and instability; instability index formula and exponential trichotomy for linearized Hamiltonian PDE; the construction of invariant manifolds near unstable coherent states. If time permits, we will also discuss long time dynamics problems such as Landau damping, inviscid damping, enhanced dissipation and asymptotic stability of solitary waves. No previous knowledge of finite dimensional Hamiltonian systems and the PDE models are required.