  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.
