Radiation damping of the soliton internal mode in 1D quadratic Klein-Gordon equation

TL;DR

This paper investigates how internal modes of solitons in a 1D quadratic Klein-Gordon equation decay over time due to radiation damping, providing a detailed quantitative analysis of energy transfer mechanisms.

Contribution

It introduces a cubic resonant approximation to accurately describe the slow decay and frequency shift of the internal mode, with damping rate determined by a Fermi Golden rule-type coefficient.

Findings

Internal mode decays slowly via radiation damping.

Energy transfer from internal mode to dispersive waves is quantitatively characterized.

Decay rate is linked to a Fermi Golden rule-type coefficient.

Abstract

We study long-time dynamics of small even perturbations of the soliton in 1D quadratic Klein-Gordon equation. The soliton possesses both an internal mode and the unstable mode. On a codimension-one manifold of fine-tuned initial data the instability is suppressed and the internal mode decays slowly by transferring energy into the continuum. We show that this decay and the associated nonlinear frequency shift are accurately captured by a cubic resonant approximation, with the damping rate determined by a Fermi golde rule-type coefficient. This provides a quantitative description of irreversible energy transfer from the internal mode to dispersive radiation.