Global existence for coupled Klein-Gordon equations with different speeds

TL;DR

This paper proves the global existence and scattering of small, smooth, localized solutions for a system of coupled Klein-Gordon equations with different speeds in three dimensions, using a novel space-time resonance analysis.

Contribution

It introduces a new analysis of the resonant structure in coupled Klein-Gordon systems with different speeds, demonstrating global solutions for small initial data.

Findings

Global existence for small data

Solutions scatter over time

Resonant structure has unique, previously unstudied features

Abstract

Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.