Long-time existence and resonant approximation for the quadratic nonlinear wave equation with an anisotropic harmonic trapping

We establish long-time existence and uniqueness for the 2D wave equation with a harmonic potential in one direction. This proof relies on a fine study of the so-called space-time resonances of the equation. Then we derive a resonant system for this equation and we prove that it is a satisfying approximation for the original equation.