The Continuing Story of the Wobbling Kink

TL;DR

This paper develops an asymptotic theory for the wobbling kink in the $^4$ model, explaining its longevity, radiation damping, and how external or parametric driving can sustain or induce motion.

Contribution

It introduces a novel asymptotic construction accounting for multiple scales and proposes mechanisms for controlling the wobbling kink via resonant driving.

Findings

Parametric pumping sustains maximum wobbling amplitude.

External driving induces a rack-and-pinion motion of the kink.

Wider Arnold tongue observed for parametric resonance.

Abstract

The wobbling kink is the soliton of the $\phi^4$ model with an excited internal mode. We outline an asymptotic construction of this particle-like solution that takes into account the coexistence of several space and time scales. The breakdown of the asymptotic expansion at large distances is prevented by introducing the long-range variables "untied" from the short-range oscillations. We formulate a quantitative theory for the fading of the kink's wobbling due to the second-harmonic radiation, explain the wobbling mode's longevity and discuss ways to compensate the radiation losses. The compensation is achieved by the spatially uniform driving of the kink, external or parametric, at a variety of resonant frequencies. For the given value of the driving strength, the largest amplitude of the kink's oscillations is sustained by the {\it parametric\/} pumping --- at its natural wobbling frequency. This type of forcing also produces the widest Arnold tongue in the "driving strength versus driving frequency" parameter plane. As for the {\it external\/} driver with the same frequency, it brings about an interesting rack and pinion mechanism that converts the energy of external oscillation to the translational motion of the kink.