4-wave dynamics in kinetic wave turbulence

Abstract

h i g h l i g h t s• 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. Keywords: Weak wave turbulence Kinetic theory Intermittency a b s t r a c tA 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.

Wave condensation with weak disorder versus beam self-cleaning in multimode fibers

“…We remark in this respect that the key assumption underlying the wave turbulence approach is the existence of a random phase among the modes rather than a genuine Gaussian statistics, as recently discussed in detail in Refs. [7,86,87]. We emphasize that in the present work the random phase of the modes is induced by the structural disorder of the medium that dominates nonlinear effects (L d L nl ).…”

“…Also note that we have deliberately chosen a small value of the condensate fraction n 0 /N 0.2 so as to avoid large deviations from Gaussianity for the fundamental mode -though the the- ory has been validated even for large condensate fractions in [41]. We recall here the recent works showing that a key assumption of the wave turbulence approach is the existence of a random phase among the modes rather than a genuine Gaussian statistics [86,87].…”

“…This rapid condensation can be explained by the perturbation on the dispersion relation due to the truncation of the parabolic potential, which modifies the regularization of wave resonances by removing the mode degeneracies, L pert kin ∼ b p L 2 nl /S 2 lmnp , see Appendix B. However, at variance with experiments of beam cleaning, this estimation of L pert kin assumes that the initial condition exhibits random phases among the modes, a feature of fundamental importance for the applicability of the wave turbulence theory [7,86,87]. In addition, this estimation assumes that the perturbation b p is maintained fixed, while in a real experiment such a perturbation may not be controlled all over the fiber length due to inherent imperfections and external perturbations, and the corresponding fluctuations contribute to the disorder term in the NLS equation, as discussed through the kinetic Eq.…”

On the properties of energy flux in wave turbulence

“…In more recent works, however, it has been shown that the quasi-Gaussian assumption can be dropped, and the WKE can be derived instead assuming random phases and amplitudes (Choi, Lvov & Nazarenko 2005 a ; Choi et al. 2005 b ; Eyink & Shi 2012; Chibbaro, Dematteis & Rondoni 2018). Depending on different methods of the derivation, takes the form of (Zakharov et al.…”

Equilibrium and nonequilibrium description of negative temperature states in a one-dimensional lattice using a wave kinetic approach

“…is the wave action spectral density, and • is performed over the initial random phases and independent actions. Strictly speaking, the WK equation is valid under the assumption that random phases and amplitudes persist over time [41]. Besides the evolution equation for the spectral density function, using the same approximations and tools, it is also possible to derive an evolution equation for the second moment, Λ k = I 2 k (M/2π) 2 , which reads:…”