Perturbation theory for localized solutions of sine-Gordon equation: decay of a breather and pinning by microresistor

TL;DR

This paper develops a perturbation theory for localized soliton solutions of the sine-Gordon equation, analyzing effects of inhomogeneities and dissipation on fluxons and breathers, with numerical validation and comparison to existing theories.

Contribution

It introduces a new perturbation framework for localized solutions of the sine-Gordon equation, accounting for inhomogeneities and dissipation, and compares numerical results with McLaughlin-Scott theory.

Findings

Microresistor can pin fluxons effectively.

Breather decay is significantly influenced by history dependence.

Numerical results differ from McLaughlin-Scott predictions, especially for small damping.

Abstract

We develop a perturbation theory that describes bound states of solitons localized in a confined area. External forces and influence of inhomogeneities are taken into account as perturbations to exact solutions of the sine-Gordon equation. We have investigated two special cases of fluxon trapped by a microresistor and decay of a breather under dissipation. Also, we have carried out numerical simulations with dissipative sine-Gordon equation and made comparison with the McLaughlin-Scott theory. Significant distinction between the McLaughlin-Scott calculation for a breather decay and our numerical result indicates that the history dependence of the breather evolution can not be neglected even for small damping parameter.