Some explicit travelling-wave solutions of a perturbed sine-Gordon equation

We present in closed form some special travelling-wave solutions (on the real line or on the circle) of a perturbed sine-Gordon equation. The perturbation of the equation consists of a constant forcing term γ and a linear dissipative term, and the equation is used to describe the Josephson effect in the theory of superconductors and other remarkable physical phenomena. We determine all travelling-wave solutions with unit velocity (in dimensionless units). For |γ| ≤ 1 we find families of solutions that are all (except the obvious constant one) manifestly unstable, whereas for |γ|> 1 we find families of stable solutions describing each an array of evenly spaced kinks. Preprint 07-09 Dip. Matematica e Applicazioni, Università di Napoli. To appear in the proceedings of the conference “Mathematical Physics Models and Engineering Sciences”, Naples 22-23 June 2006, in honour of P. Renno’s for his 70 birthday. 1