On the onset of correlations in Wave Turbulence close to singularities

TL;DR

This paper analyzes the failure of the wave turbulence kinetic equation near singularities in the Schrödinger equation, proposing a new hierarchy of equations involving a nonlinear non-autonomous Schrödinger equation.

Contribution

It introduces a novel hierarchy of equations to describe wave turbulence near blow-up times, extending beyond the traditional kinetic equation approach.

Findings

Kinetic equation derivation breaks down near singularities.

A hierarchy of equations equivalent to a nonlinear Schrödinger equation is needed.

Near blow-up, wave turbulence behavior is governed by a different mathematical framework.

Abstract

In this paper we describe in a formal way how the derivation of the turbulent wave equation for the Schr\"odinger equation breaks down for times close to the self similar blow up of the wave turbulence kinetic equation. To this end, we study how the derivation of the cumulants hierarchy can not be approximated using solutions of the wave turbulence kinetic equation near the blow up time. It tuns out that near the blow up time the kinetic equation has to be replaced by a hierarchy of equations which is equivalent to a random field, defined for times $t\in (-\infty, \infty)$ and satisfying a nonlinear non autonomous Schr\"odinger equation.