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

TL;DR

This paper proves long-time existence and uniqueness for a 2D nonlinear wave equation with anisotropic harmonic trapping, using space-time resonance analysis, and shows that a derived resonant system accurately approximates the original equation.

Contribution

It introduces a detailed resonance analysis for the anisotropic wave equation and establishes the validity of a resonant system as an approximation.

Findings

Long-time existence and uniqueness are proven for the 2D wave equation with harmonic potential.

A resonant system is derived and shown to approximate the original equation effectively.

The study advances understanding of resonant dynamics in anisotropic nonlinear wave equations.

Abstract

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.