Abstract: Weak turbulence theory is a kinetic theory of wave turbulence. Recently, there are many new theoretical progresses. Here, we discuss a theoretical framework that allows us to treat the non-perturbative nature of wave turbulence in strongly nonlinear regmies. We show how the wave spectrum n(k) of nonlinear dispersive waves is determined by an intertwining self-consistent process: The trivial resonant scatterings of waves off of background waves characterized by n(k) control the true, renormalized, dispersion relation. The renormalized dispersion relation, in turn, controls nontrivial resonances of the full wave system, thus, giving rise to a self-consistent wave spectrum n(k). We present an extension of the weak turbulence kinetic theory to systems with strong nonlinearities. |