Open-channel radiation zeros and nonlinear damping of a critical-bubble internal mode

Abstract

We study radiation and nonlinear damping of a spherically symmetric localized mode of an $O(3)$ thermal critical bubble. The leading second-harmonic radiation amplitude is determined by the overlap between a nonlinear source and a continuum scattering wave. This overlap can vanish through destructive interference as the supercooling is varied, even when the second harmonic lies above the continuum threshold. A signed-overlap scan resolves five zeros in a finite parameter interval, including three on the thinner-wall side of the two representative roots studied in detail. For these representative roots, half-line Jost--Green calculations, threshold-preserving wall deformations, and constrained radial evolution verify the cancellation and its robustness. Near a selected root, fourth-order perturbation theory predicts an $A^2$ displacement of the finite-amplitude radiation minimum and an $A$-linear width of the third-harmonic-dominated region; radial evolution quantitatively tests both relations. At the selected zero, third-harmonic loss predicts $A\simτ^{-1/4}$, instead of the generic $A\simτ^{-1/2}$ law. Measured harmonic powers and a slow-envelope reconstruction support this hierarchy. Independent center-stable shooting recovers the unprojected third-harmonic coefficient, which differs from the fixed-projector value. The results connect a tunable radiation form factor to nonlinear relaxation of an internal mode on an unstable nucleation saddle.