A
large number of partial differential equations of Physics have the
structure of an infinite-dimensional Hamiltonian dynamical system. In
this class of equations appear, among others, the Schrödinger equation
(NLS), the wave equation (NLW), the Euler equations of hydrodynamics
and the numerous models that derive from it. The study of these
equations poses some fundamental questions that have inspired an entire
research field in the last twenty years: the investigation of the main
invariant structures of the phase space of a Hamiltonian system,
starting from its stationary, periodic and quasi-periodic orbits. As in
the case of finite-dimensional dynamical systems, one of the main
problems in this field is linked to the well-known "small divisors
problem''. A further difficulty is due to the fact that
''physically'' interesting equations, without outer parameters,
are typically resonant and/or contain derivatives in the non-linearity.
There are many fundamental open questions in this field. Our main goals
are 1) the study of quasi-periodic solutions, in particular for
semi-linear and quasi-linear equations 2) Study of normal forms, both
in integrable and non-integrable cases. 3) Applications to
hydrodynamics and search of quasi-periodic solutions in water wave
problems. 4) Study of almost periodic solutions for semilinear PDEs. 5)
Quasi-periodic solutions for resonant systems (both finite and
infinite dimensional) with minimal restrictions on the non-linearity.
Together with my group in Naples we have already
developed several techniques to approach these problems and we have
several ideas of possible innovative approaches, combining Nash-Moser
and KAM methods, Normal Form Theory, Para-differential calculus,
combinatorial methods.
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