Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities

TL;DR

This paper studies the long-term behavior of small solutions to 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities, revealing resonant interactions and modified scattering phenomena.

Contribution

It introduces a new analysis of resonant interactions and a novel variable coefficient quadratic normal form for Klein-Gordon equations with variable nonlinearities.

Findings

Resonant interaction causes logarithmic slow-down in decay.

Sharp decay estimates are established for non-resonant cases.

Modified scattering behavior is characterized along certain rays.

Abstract

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.