  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.
