Mathematics

Invariant manifolds are fundamental tools in the study of dynamical systems generated by differential equations. They provide coordinates in which the systems can be partially decoupled and can be used to track the asymptotic behaviors of the orbits. Therefore, starting with Poincare, Hadamard, Lyapunov, Perron and et al., people have studied extensively their existence, smoothness, and persistence under small perturbations (such as those due to the modelling procedure,  small noises, or computational round-off error, etc.). In nonlinear PDE  dynamics, issues such as spatial regularity and multiple spatial and temporal scales bring extra challenges. In these lectures, we will:

1. first review some basic invariant manifold theory for semilinear PDEs,
2. hen study invariant manifolds for a class of quasilinear PDEs, and then
3. discuss invariant manifolds and applications in several concrete PDE dynamical systems possibly including the nonlinear Klein-Gordon equation, the nonlinear Schrodinger equation, the Euler equation, and the spike  dynamics of a singular parabolic equation, etc. 

Dates 17-31 July 2025