Kinks in the Presence of Rapidly Varying Perturbations

TL;DR

This paper develops an analytical and numerical framework to understand sine-Gordon kink dynamics under rapidly varying periodic perturbations, revealing effective equations and new phenomena in driven nonlinear systems.

Contribution

It introduces a rigorous asymptotic method to derive effective nonlinear equations for sine-Gordon kinks under high-frequency perturbations, including renormalized and double sine-Gordon equations.

Findings

Effective equations are derived for different types of driving forces.

New phenomena such as perturbation-induced effects are predicted.

Numerical simulations confirm analytical predictions.

Abstract

Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows to derive, in a rigorous way, an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the small parameter $\omega^{-1}$, $\omega$ being the frequency of the rapidly varying ac driving force. Three physically important examples of such a dynamics, {\em i.e.}, kinks driven by a direct or parametric ac force, and kinks on rotating and oscillating background, are analysed in detail. It is shown that in the main order of the asymptotic procedure the effective equation for the slowly varying field component is {\em a renormalized sine-Gordon equation} in the case of the direct driving force or rotating (but phase-locked to an external ac force) background, and it is {\em the double sine-Gordon equation} for the parametric driving force. The properties of the kinks described by the renormalized nonlinear equations are analysed, and it is demonstrated analytically and numerically which kinds of physical phenomena may be expected in dealing with the renormalized, rather than the unrenormalized, nonlinear dynamics. In particular, we predict several qualitatively new effects which include, {\em e.g.}, the perturbation-induced