Parametric Resonance in the Non-Autonomous Sine-Gordon Model

TL;DR

This paper investigates parametric resonance phenomena in a non-autonomous sine-Gordon model with space-time dependent parameters, using an effective reduced model validated against full field simulations.

Contribution

It develops a simplified effective model for the sine-Gordon system under temporal and spatial inhomogeneity and validates it against full field model results.

Findings

Good agreement between effective and full models in stability regions.

Identified the boundaries of Arnold tongues in the parametric instability.

Observed more complex dynamics in the full model at the bottom of Arnold's tongues.

Abstract

The sine-Gordon model is studied with model parameters that depend on both space and time. An effective model with one degree of freedom is constructed, allowing the description of the kink movement in both a temporally non-autonomous and spatially inhomogeneous setting. As a highly demanding test of the reduced model, the case of a temporal drive leading to a parametric instability is considered. Here, the boundaries of the instability region in the form of Arnold tongues were examined in both the simplified effective model and the full field model. Good agreement was obtained as concerns the regions of stability. It was observed that only in the bottom parts of Arnold's tongues is the dynamics of the field model more complicated than in the approximate model.