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

Final Report Summary - HAMPDES (Hamiltonian PDE's and small divisor problems: a dynamical systems approach.)

The general field of interest of this project is the study of classes Partial Differential Equations on compact domains, which model the behaviour of waves. In this class we mention among others, the nonlinear Schrodinger equation (NLS), the nonlinear wave equation (NLW), the Euler equations of hydrodynamics and the numerous models that derive from it.
Our point of view in attacking this vast problem is to study special global solutions such as elliptic fixed points or periodic, quasi-periodic and almost periodic solutions. The existence of such solutions is interesting per se, but a main question is to study their stability (and instability), which gives information on the evolution of near-by initial data.
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.
The methods used to overcome these difficulties are KAM /Nash-Moser quadratic algorithms, pseudo and para differential calculus and Birkhoff normal forms, combined with algebro/geometric ideas used to deal with resonances.
Our main achievements were: the study of quasi-periodic solutions for various PDEs on the circle. In particular for semi-linear, quasi-linear and fully non-linear equations we proposed a very effective strategy which enabled us to prove existence and linear stability. A particular success was the application to the Water Wave problem.
Study of normal forms, both in integrable and non-integrable cases. We proposed a strategy for dealing with resonant PDEs in high spacial dimension which allowed us to prove linear stability for quasi-periodic solutions of the NLS on any torus, and conversely non-linear instability for the same model close one dimensional finite gap solutions. A novel, more geometric approach to the costruction of almost periodic solutions for semilinear PDEs with external parameters.