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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

Linear and nonlinear bifurcation problems

After introducing the theory of analytic functions between Banach spaces,  we shall present perturbative results for the spectrum of linear operators, in particular for separated eigenvalues of closed operators, with applications to the stability of traveling water waves. Then we shall present  bifurcation results  of periodic and quasi-periodic solutions of nonlinear dynamical systems as well as homoclinic solutions to hyperbolic equilibria of Hamiltonian systems.

Water waves

The water waves equations were introduced by Euler in the 18th century to describe the motion of a mass of water under the influence of gravity and with a free surface. The unknown of the problem are two time dependent functions describing how the velocity field of the fluid and the profile of the free surface (giving the shape of the waves) evolve. The mathematical analysis of the water waves equations is particularly challenging due to their quasilinear nature, and it has been (and still is) a central research line in fluid dynamics.

Periodic Orbits of Hamiltonian systems through Variational Methods

Hamiltonian systems give a very good description of those physical phenomena where the energy is (approximately) conserved: from planetary orbits to the motion of particles.

Typically, however, the dynamics is highly sensitive to the initial conditions and therefore it is difficult to find specific orbits in the systems such as those connecting two subsets of phase space or those which are periodic.

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


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