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

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.