| The aim of this thesis is to go beyond the traditional theoretical characterizations of turbulence in nonlinear dispersive waves. It consists of two more or less independent parts.;In the first part, using the Majda-McLaughlin-Tabak model and the generalized Fermi-Pasta-Ulam chains as two illustrative examples, we present an extension of the Wave Turbulence theory to systems with strong nonlinearities. It is demonstrated that nonlinear wave interactions renormalize the dynamics, leading to (i) a possible destruction of scaling structures in the bare wave systems and a drastic deformation of the resonant manifold even at weak nonlinearities, and (ii) an effective oscillation in the systems for which no bare wave dynamics exists and creation of nonlinear resonance quartets in wave systems for which there would be no resonances as predicted by the linear dispersion relation. We derive an effective kinetic equation and show that our prediction of the renormalized Rayleigh-Jeans distribution is in excellent agreement with the simulation of the full wave system in equilibrium.;In the second part, a new model for studying energy transfer among waves is introduced. It consists of hierarchical four-wave resonant quartets coupled by one-mode which is in an inertial range, while the rest of waves would be simultaneously forced by white noise and damped with distinct strength. In this driven-damped system, the equilibrium measure is given by a random phase, independent normal distribution satisfying energy balance condition. Furthermore, an analytic closure is derived to predict the nonequilibrium statistics of the weakly interacting system. The relation of the flux dynamics to entropy maximum principle will be discussed. |
