Related papers: Long time justification of wave turbulence theory
We examine the validity of the kinetic description of wave turbulence for a model quadratic equation. We focus on the space-inhomogeneous case, which had not been treated earlier; the space-homogeneous case is a simple variant. We determine…
A fundamental question in wave turbulence theory is to understand how the "wave kinetic equation" (WKE) describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature date back…
We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schr\"odinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main…
We introduce a simplified model for wave turbulence theory -- the Wick NLS, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic…
Wave turbulence is the study of the long-time statistical behaviour of equations describing a set of weakly non-linear interacting waves. Such a theory, which has a natural asymptotic closure, allows us to probe the nature of turbulence…
Using the 1D Majda-McLaughlin-Tabak model as an example, we develop numerical experiments to study the validity of the Wave Kinetic Equation (WKE) at the kinetic limit (i.e., small nonlinearity and large domain). We show that the dynamics…
We present a systematic derivation of the wave kinetic equation describing the dynamics of a statistically inhomogeneous incoherent wave field in a medium with a weak quadratic nonlinearity. The medium can be nonstationary and…
In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation (WKE). This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and…
Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKE) for water waves models. This problem has received intense attention in recent years in the context of…
We test the predictions of the theory of weak wave turbulence by performing numerical simulations of the Gross-Pitaevskii equation (GPE) and the associated wave-kinetic equation (WKE). We consider an initial state localized in Fourier…
Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the…
The main purpose of this expository note is to give a short account of the recent developments in mathematical wave kinetic theory. After reviewing the physical theory, we explain the importance of the notion of a scaling law, which…
Many new models of wave turbulence -- frozen, mesoscopic, laminated, decaying, sand-pile, etc. -- have been developed in the last decade aiming to solve problems seemingly not solvable in the framework of the existing wave turbulence theory…
We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck…
The results from J. Stat. Phys. 159:668-712 & 163:1350-1393, on a quadratic kinetic equation in the analysis of the long time asymptotics of weak turbulence theory for the nonlinear Schr\"odinger equation, are summarized and placed in…
We provide a rigorous derivation of Boltzmann's kinetic equation from the hard sphere system for rarefied gas, which is valid for arbitrarily long times, as long as the solution to the Boltzmann equation exists. This extends Lanford's…
We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.
The recent observation of gravitational waves, stimulates the question of the longtime evolution of the space-time fluctuations. Gravitational waves interact themselves through the nonlinear character of Einstein's equations of general…
The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectrum in weakly nonlinear and dispersive media. Such a kinetic model has been derived by physicists in the sixties, though the…
In wave turbulence, it has been believed that statistical properties are well described by the weak turbulence theory, in which nonlinear interactions among wavenumbers are assumed to be small. In the weak turbulence theory, separation of…