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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

Marie Curie Research and Innovation Staff Exchange, "Integrable Partial Differential Equations: Geometry, Asymptotics, and Numerics"

IPaDEGAN is a European Marie Skłodowska-Curie Research and Innovation Staff Exchange ( RISE ) project, funded by the European Commission within the H2020-MSCA-RISE-2017 call. It fosters international mobility and collaboration on the topic of partial differential equations, especially on Integrable PDEs and their ramified applications.

Pseudodifferential operators, applications and dynamics

Aim of the course is to introduce the basic tools of pseudodifferential calculus, and to show applications of such techniques in the analysis of dispersive PDEs, spectral theory, or other areas. A particular emphasis will be given to the problem of growth of Sobolev norms. 

Course Contents:

Part 1: Review of Fourier calculus

Bifurcation theory and PDEs

This course deals with bifurcation theory and applications to dynamical systems and PDEs, like the Lyapunov center theorem, Hopf bifurcation, traveling and Stokes waves for fluids. At the beginning we shall present the differential calculus and the implicit function theorem in Banach spaces. At the end I will deal also with the cases in which the classical implicit function theorem can not be applied since the linearized operator has an unbounded inverse and a version of the Nash-Moser implicit function theorem. 

Introduction to pseudodifferential operators and dynamics of linear, time dependent Schrödinger equations

Aim of the course is to introduce the basic tools of pseudodifferential calculus, and apply them to analyze the long time dynamics of linear, time dependent Schrödinger equations. A particular emphasis will be given to the problem of growth of Sobolev norms.

Course contents:

Part 1: Pseudodifferential operators

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.


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