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Dynamical Systems and PDEs

  • KAM for PDEs
    • Periodic and Quasi-periodic solutions of Hamiltonian PDEs
    • Nonlinear wave and Schroedinger equations
    • Reversible KAM theory
    • KAM for unbounded perturbations: quasi linear KdV, derivative wave equations
    • Water waves equations
    • Birkhoff Lewis periodic orbits
    • Almost periodic solutions
  • Bifurcation Theory and Nash-Moser Implicit Function Theorems
  • Birkhoff normal forms
  • Variational and Topological Methods in the study of Hamiltonian systems
    • Variational methods for periodic solutions
    • Homoclinic and heteteoclinic solutions
  • Dynamical systems
    • Arnold Diffusion
    • Chaotic dynamics
    • Perturbation and Nekhoroshev Theory
    • 3 body problem

Reducibility and KAM theory in PDEs

The main focus of the course will be on the problem of reducibility of linear time dependent PDEs, namely the problem of finding a coordinate transformation conjugating the equation to a time independent one. The course can be considered as an introduction to KAM theory and its use in PDEs. I will start by presenting a related topic, namely Poincare theory for the persistence of periodic orbits, which is one of the theory in which elementary reducibility theory finds application.

Dynamics of nonlinear PDEs

We shall discuss the complex dynamics of Hamiltonian or reversible PDEs, like the NLS,  the Klein-Gordon equation, the wave equation in several dimension,the water waves equations of hydrodynamics, focusing on both chatotic and regular behaviour as the existence of quasi-periodic solutions.We shall discuss also long time existence results obtained by Birkhoff normal forms.

Dynamical Systems and PDEs

Research topics

  • KAM for PDEs
  • Water waves, KdV, Schrodinger and Klein-Gordon equations
  • Hamiltonian systems
  • Birkhoff normal form for PDEs and long time existence
  • Chaotic Dynamics and Arnold Diffusion
  • Variational Methods
  • Perturbative Methods in Critical Point Theory
  • Elliptic Equations on $\mathbb R^n$ and Nonlinear Schrodinger Equation

Research Group

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