4-wave dynamics in kinetic wave turbulence

A general Hamiltonian wave system with quartic resonances is considered, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. The evolution equation for the multimode characteristic function Z is obtained within an “interaction representation” and a perturbation expansion in the small nonlinearity parameter. A frequency renormalization is performed to remove linear terms that do not appear in the 3-wave case. Feynman–Wyld diagrams are used to average over phases, leading to a first order differential evolution equation for Z. A hierarchy of equations, analogous to the Boltzmann hierarchy for low density gases is derived, which preserves in time the property of random phases and amplitudes. This amounts to a general formalism for both the N-mode and the 1-mode PDF equations for 4-wave turbulent systems, suitable for numerical simulations and for investigating intermittency. Some of the main results which are developed here in detail have been tested numerically in a recent work. •We deal with 4-wave Hamiltonian systems in the framework of wave turbulence.•Averaging technique based on the Feynman–Wyld diagrams.•Kinetic limit : leading order equations for the statistics evolution are derived.•Random-phase and random-phase and amplitude properties preserved in time.•Powerful tool to investigate many relevant physical systems.