Exact analytical solution of the problem of current-carrying states of the Josephson junction in external magnetic fields

The classical problem of the Josephson junction of arbitrary length $W$ in the presence of externally applied magnetic fields $(H)$ and transport currents $(J)$ is reconsidered from the point of view of stability theory. In particular, we derive the complete infinite set of exact analytical solutions for the phase difference that describe the current-carrying states of the junction with arbitrary $W$ and an arbitrary mode of the injection of $J$. These solutions are parametrized by two natural parameters: the constants of integration. The boundaries of their stability regions in the parametric plane are determined by a corresponding infinite set of exact functional equations. Being mapped to the physical plane $(H,J)$, these boundaries yield the dependence of the critical transport current ${J}_{c}$ on $H$. Contrary to a widespread belief, the exact analytical dependence ${J}_{c}={J}_{c}(H)$ proves to be multivalued even for arbitrarily small $W$. What is more, the exact solution reveals the existence of unquantized Josephson vortices carrying fractional flux and located near one of the junction edges, provided that $J$ is sufficiently close to ${J}_{c}$ for certain finite values of $H$. This conclusion (as well as other exact analytical results) is illustrated by a graphical analysis of typical cases.

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