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Hamiltonian PDE's and small divisor problems: a dynamical systems approach

Objective

"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 nonlinear PDEs. 5) quasi-periodic solutions for resonant systems with minimal restrictions on the non-linearity. Together with my group in Naples we already have 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."

Call for proposal

ERC-2012-StG_20111012
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Host institution

UNIVERSITA DEGLI STUDI ROMA TRE
Address
Via Ostiense 133
00154 Roma
Italy
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 297 772,46
Principal investigator
Michela Procesi (Prof.)
Administrative Contact
Virgilio Lo Presti (Mr.)

Beneficiaries (3)

UNIVERSITA DEGLI STUDI ROMA TRE
Italy
EU contribution
€ 297 772,46
Address
Via Ostiense 133
00154 Roma
Activity type
Higher or Secondary Education Establishments
Principal investigator
Michela Procesi (Prof.)
Administrative Contact
Virgilio Lo Presti (Mr.)
UNIVERSITA DEGLI STUDI DI NAPOLI FEDERICO II
Italy
EU contribution
€ 215 272,56
Address
Corso Umberto I, 40
80138 Napoli
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Carmela Ippolito (Ms.)
UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA
Italy
EU contribution
€ 164 954,98
Address
Piazzale Aldo Moro 5
00185 Roma
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Maria Rita Ullucci (Ms.)