Instability of singular equilibria of a wave kinetic equation

TL;DR

This paper investigates the instability of singular equilibria in a wave kinetic equation related to 4-wave turbulence, demonstrating finite-time formation of a Dirac measure and analyzing long-term convergence behavior.

Contribution

It provides the first analysis of finite-time singularity formation and long-term convergence to a Dirac measure in the context of wave kinetic equations for 4-wave turbulence.

Findings

Finite-time formation of a Dirac measure at zero frequency.

Linearization around singular equilibria is well-posed in various spaces.

Long-term convergence to the Dirac measure at the origin for certain solutions.

Abstract

We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic turbulence equation for the three dimensional Schr\"{o}dinger equation. We first show the formation in finite time of a Dirac measure at zero frequency in the solution of the wave kinetic equation when the initial data has the form of Rayleigh-Jeans, truncated at large values of the energy. The initial value problem for the linearization around the singular Rayleigh-Jeans equilibria is then solved in several functional spaces. Then, long time convergence to a Dirac measure at the origin is described in detail for some of the solutions. This determines a basin of attraction of the Dirac measure.