Radiation damping of the soliton internal mode in one-dimensional quadratic Klein-Gordon equation

We study the long-time dynamics of small even perturbations of the soliton in a one-dimensional quadratic nonlinear Klein-Gordon equation. The soliton possesses both an internal mode and an 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 golden rule-type coefficient. This provides a quantitative description of an irreversible energy transfer from the internal mode to dispersive radiation.