Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities

TL;DR

This paper investigates the scattering behavior of solutions to the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities, revealing how resonances influence long-term dynamics and introducing new normal-form transformations.

Contribution

It introduces novel normal-form transformations to analyze resonant interactions in the Klein-Gordon equation with variable nonlinearities, providing new insights into scattering and solution smoothness.

Findings

Proved L-infinity scattering in non-resonant cases

Established strong smoothness of solutions at time-like infinity

Identified qualitatively different behaviors in resonant cases

Abstract

We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case where the worst of this resonant behavior is absent, we prove L-Infinity scattering as well as a certain kind of strong smoothness for the solution at time-like infinity with the help of several new normal-forms transformations. Some explicit examples are also given which suggest qualitatively different behavior in the case where the strongest cubic resonances are present.