On kinks and other travelling-wave solutions of a modified sine-Gordon equation

TL;DR

This paper classifies all exact travelling-wave solutions of a perturbed sine-Gordon equation used in superconductivity, identifying new solution types and proposing a convergent approximation method for kink solutions.

Contribution

It provides a comprehensive, non-perturbative classification of solutions, including a novel 'half-array-of-kinks' type, and introduces a fixed point theorem-based method for approximating kink solutions.

Findings

Identification of constant, kink, array-of-kinks, and half-array-of-kinks solutions.

Introduction of a convergent approximation method for kink solutions.

Discovery of a new solution type not present in unperturbed equations.

Abstract

We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of superconductors and other remarkable physical phenomena. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. On the real line candidate orbitally stable solutions with bounded energy density are either the constant one, or of kink (i.e. soliton) type, or of array-of-kinks type, or of "half-array-of-kinks" type. While the first three have unperturbed analogs, the last type is essentially new. We also propose a convergent method of successive approximations of the (anti)kink solution based on a careful application of the fixed point theorem.