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

Research Group:

Mathematical Analysis, Modelling, and Applications

Fast oscillating forced systems: reducibility for a linear Klein-Gordon equation

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Almost global solutions for the quasi-linear Schrödinger equation with small periodic data

Read more about Almost global solutions for the quasi-linear Schrödinger equation with small periodic data

A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs

Read more about A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs

Quasi-periodic solutions for quasi-linear generalized KdV equations

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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.

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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.

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