Wave turbulence for a semilinear Klein-Gordon system

In this article we consider a system of two Klein-Gordon equations, set on the $d$-dimensional box of size $L$, coupled through quadratic semilinear terms of strength $\varepsilon$ and evolving from well-prepared random initial data. We rigorously derive the effective dynamics for the correlations associated to the solution, in the limit where $L\to\infty$ and $\varepsilon\to 0$ according to some power law. The main novelty of our work is that, due to the absence of invariances, trivial resonances always take precedence over quasi-resonances. The derivation of the nonlinear effective dynamics is justified up time to $\delta T$ , where $T =\varepsilon^{-2}$ is the appropriate timescale and $\delta$ is independent of $L$ and $\varepsilon$. We use Feynmann interaction diagrams, here adapted to a normal form reduction and to the coupled nature of our real-valued system. We also introduce a frequency decomposition at the level of the diagrammatic and develop a new combinatorial tool which allows us to work with the Klein-Gordon dispersion relation.