Exact analytical solution of a classical Josephson tunnel junction problem

Abstract

We give an exact and complete analytical solution of the classical problem of a Josephson tunnel junction of arbitrary length W∊(0,∞) in the presence of external magnetic fields and transport currents. Contrary to a wide-spread belief, the exact analytical solution unambiguously proves that there is no qualitative difference between so-called “small” (W⪡1) and “large” junctions (W⪢1). Another unexpected physical implication of the exact analytical solution is the existence (in the current-carrying state) of unquantized Josephson vortices carrying fractional flux and located near one of the edges of the junction. We also refine the mathematical definition of critical transport current.