Dinamica delle Interazioni a 4-onde in Turbolenza d'Onda

Wave turbulence concerns the study of dispersive nonlinear waves. Generally speaking, the theory is statistically described via the spectrum, defined as the Fourier transform of the 2-point correlation function and thus linked to the second momentum of the probability density function (PDF). In particular, in order to derive a ``closed" kinetic equation for the spectrum, a simplifying hypothesis of quasi-Gaussian distribution has to be made. However, this rules out the possibility of considering strong deviations from Gaussianity and thus the prediction of extreme spontaneous events is excluded (e.g. "rogue waves" for the ocean surface gravity waves). Still, it remains important to predict such rare events in the tails of the distribution and so one has to go beyond the spectrum description. An early attempt was represented by the Peierls equation for the PDF (1929), then derived and generalized in recent works surveyed in the Nazarenko's book of 2011. However, this equation was never used sistematically, also because quite difficult to handle. In 2012, G. E. Eyink and Y. K. Shi argued that Peierls equation is not the correct asymptotic equation of the wave turbulence problem in the large-box limit. They showed that a correct normalization of amplitudes leads to simpler equations with important properties for the multimode statistics of a 3-wave resonant system. On the other hand, it is known that the most relevant physical systems are the 4-wave resonant ones. In this perspective, the aim of this thesis is to extend their new idea to the 4-wave case. We consider a general Hamiltonian wave system with quartic resonances, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. We consider the Hamiltonian in the Fourier space (in canonical normal variables) and we write Hamilton's equations. The following steps are performed to obtain an evolution equation for the multimode characteristic function Z: an ``interaction representation" averaging over fast linear oscillations, a switch to phase-amplitude variables and a ``perturbative expansion" in the small nonlinearity parameter. We remark that a frequency renormalization has to be done to remove some additional linear terms, appearing only in the 4-wave case. Now, a Feynman-Wyld diagrams method is used to average over phases and this represents the most challenging passage in this thesis. Then, the dynamical equation becomes a simple differential equation for Z, of the first order in time. We derive a hierarchy of equations analogous to the Boltzmann hierarchy in the low density limit for gases. Those equations possess factorized solutions for factorized initial data, which correspond to preservation in time of the property of ``random phases and amplitudes" (RPA). Thus, assuming RPA property for the initial wavefield, the evolution of the statistics is determined by a simple equation for the 1-mode PDF. Future numerical analysis should be thought to study the equilibrium and nonequilibrium solutions of that equation. In conclusion, on the one hand we have developed a general formalism with which we have calculated the N-mode and the 1-mode PDF equations for the 4-wave turbulent systems. That should allow a deeper analysis of the underlying hypothesis of wave turbulence. Moreover, those equations appear suitable for numerical simulations and thus we expect that they may somehow shed light on the problem of wave turbulence intermittency.