Discrete wave turbulence for a coupled system of quintic Schrödinger equations

We derive rigorously the non-linear macroscopic system associated to a microscopic system of coupled quintic Schrödinger equations in the framework of discrete wave turbulence under a particular scaling law that describes the limiting process. Our system evolves from a pair of well-prepared random initial data. More precisely, in dimensions $d\ge 2$, we set up our microscopic system on a large box of size $L$ with weak non-linearity of strength $\epsilon$. In the limit $L\to \mathrm{\infty }$ and $\epsilon \to 0$, under the scaling law $\epsilon L^{\frac{1}{\beta }}=1$ with $\beta \in (1,\mathrm{\infty })$, we prove that the long-time behaviour of our microscopic system is statistically described up to times $\delta {\epsilon }^{-1}$ by a non-linear resonant system whose dynamics are driven by exact resonances, where $\delta$ is independent of $L$ and $\epsilon$. Our system does not display generic symmetries, in particular not mass conservation. In such systems with fewer invariances, exact resonances contribute significantly compared to quasi-resonances and are essentially responsible for the effective dynamics in the large-box limit. We justify the emergence of discrete wave turbulence for our microscopic model.