Singularities in the weak turbulence regime for the quintic Schr\"odinger equation

TL;DR

This paper investigates the derivation of kinetic equations in weak turbulence for the quintic Schrödinger equation, revealing the existence of singularities where the correlation functions become discontinuous.

Contribution

It demonstrates that under certain regimes, the correlation functions exhibit an infinite number of discontinuities, highlighting singularities in the weak turbulence regime.

Findings

Correlation functions match expected physics form

Presence of infinitely many discontinuities

Singularities occur in specific regimes

Abstract

In this paper, we discuss the problem of derivation of kinetic equations from the theory of weak turbulence for the quintic Schr\"odinger equation. We study the quintic Schr\"odinger equation on $L\mathbb T$, with $L\gg 1$ and with a non-linearity of size $\varepsilon\ll 1$. We consider the correlations $f(T)$ of the Fourier coefficients of the solution at times $t = T\varepsilon^{-2}$ when $\varepsilon\rightarrow 0$ and $L\rightarrow \infty$. Our results can be summed up in the following way : there exists a regime for $\varepsilon$ and $L$ such that for $T$ dyadic, $f(T)$ has the form expected from the physics literature, but such that $f$ has an infinite number of discontinuity points.