Resonantly driven wobbling kinks

TL;DR

This paper investigates how resonant driving can sustain and control wobbling oscillations of kinks in the $\

Contribution

It derives amplitude equations for driven wobbling kinks, revealing multistability, hysteresis, and the conditions for strongest resonance, verified through numerical simulations.

Findings

Resonant driving can compensate radiation losses in wobbling kinks.

Strongest parametric resonance occurs at the natural wobbling frequency.

Direct driving at half the natural frequency induces kink motion.

Abstract

The amplitude of oscillations of the freely wobbling kink in the $\phi^4$ theory decays due to the emission of second-harmonic radiation. We study the compensation of these radiation losses (as well as additional dissipative losses) by the resonant driving of the kink. We consider both direct and parametric driving at a range of resonance frequencies. In each case, we derive the amplitude equations which describe the evolution of the amplitude of the wobbling and the kink's velocity. These equations predict multistability and hysteretic transitions in the wobbling amplitude for each driving frequency -- the conclusion verified by numerical simulations of the full partial differential equation. We show that the strongest parametric resonance occurs when the driving frequency equals the natural wobbling frequency and not double that value. For direct driving, the strongest resonance is at half the natural frequency, but there is also a weaker resonance when the driving frequency equals the natural wobbling frequency itself. We show that this resonance is accompanied by translational motion of the kink.