Wave turbulence for a semilinear Klein-Gordon system
Anne-Sophie de Suzzoni, Annalaura Stingo, Arthur Touati
TL;DR
The paper analyzes a pair of coupled quadratic Klein–Gordon equations on a large torus with random initial data and quadratic nonlinearities. After a normal-form reduction, trivial resonances, rather than quasi-resonances, govern the long-time, nonlinear evolution of correlations, leading to a nonlinear, non-kinetic effective dynamics. The authors develop a novel diagrammatic framework using signed coloured trees, couplings, and bushes, and they prove high-probability convergence of the Dyson-series expansion up to times of order $\delta\varepsilon^{-2}$ with $\varepsilon=L^{-1/\beta}$, by splitting high/low frequency interactions and carefully bounding resonant contributions. The resulting effective system for the correlations, a triangular cross-system for $\rho^0$, $\rho^1$, and $\rho^\times$, reveals how trivial resonances shape energy transfer in the discrete-box regime and contrasts with kinetic wave turbulence, providing rigorous insight into nonlinear correlation dynamics for real-valued wave systems.
Abstract
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 $δT$ , where $T =\varepsilon^{-2}$ is the appropriate timescale and $δ$ 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.