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We study the sine-Gordon equation and systems of discrete approximations to it which are respectively a model of the Josephson junction and models of coupled-point Josephson junctions. We do this in the so-called current-driven case. The voltage response of these devices is the average of the time derivative of the solution of these equations and we demonstrate the existence of running periodic solutions for which the average exists. Static solutions are also studied. These together with the running solutions explain the multiple-valuedness in the response of a Josephson junction in several cases. The stability of the various solutions is described in some of the cases. Numerical results are displayed which give details of structure of solution types in the case of a single point junction and of the one-dimensional distributed junction.
Levi et al. (1978) studied this question.