A soliton in the quadratic Klein-Gordon equation has an internal mode — a localized oscillation riding on the soliton's shape, vibrating at a frequency inside the spectral gap. It also has an unstable mode that would destroy the soliton if excited. On a carefully tuned set of initial conditions (a codimension-one manifold), the instability is suppressed. What remains is the internal mode, ringing.
The internal mode decays. Not because anything forces it to stop, but because its frequency eventually couples to the radiation continuum through nonlinear effects. Energy leaks out as dispersive waves — radiation that propagates away and never returns. The soliton gradually stills.
The rate of this leaking follows a Fermi golden rule, borrowed from quantum mechanics: a discrete state coupled to a continuum decays at a rate set by the coupling strength and the density of states at the resonant frequency. The soliton's internal vibration is the “excited atom.” The continuum of dispersive waves is the “electromagnetic vacuum.” The internal mode doesn't stop because of friction. It stops because the universe of possible wave modes accepts its energy and disperses it irreversibly.
What makes this precise is that a cubic resonant approximation captures both the decay rate and the nonlinear frequency shift. The internal mode doesn't just die — its frequency drifts as it radiates, and the drift is quantitatively predicted by the same mechanism that predicts the decay. The damping and the frequency shift are the same process viewed from different angles.
The through-claim: irreversibility in a conservative system (no friction, no dissipation) arises from coupling to a continuum. The energy isn't lost — it's distributed beyond recovery. The soliton is permanent. Its internal vibration is not.