Collective coordinate descriptions of a kink in a driven-damped $\phi^4$ model

TL;DR

This paper develops and compares effective models for describing the dynamics of $^4$ kinks under external perturbations, finding that models based on position and width are most accurate.

Contribution

It introduces three reduced models for $^4$ kink dynamics and systematically compares their accuracy against full numerical solutions.

Findings

Position and width based model best matches numerical solutions.

Moderate external driving frequency allows accurate modeling.

Effective models capture complex dynamical processes accurately.

Abstract

Extending a recent effective theory formulation for the dynamics of kinks in the sine-Gordon model [1], we propose an analogous effective description of $\phi^4$ kinks. Three different reduced models based on the kink position, width and internal mode amplitude are introduced and compared systematically with the numerical solution of the equation with space- and time-dependent perturbations. In all cases considered, the model based on the kink position and width agrees the best with the full numerical solution. As long as the external driving frequency of the perturbation remains moderate, it captures with remarkable accuracy the intricate dynamical processes taking place in the system.