**Физика Низких Температур: ***Том ***36**, Выпуск **10-11** (Октябрь 2010), c. 1253-1263
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### Exact analytical solution of a classical problem of the Josephson tunnel junction

**S.V. Kuplevakhsky and A.M. Glukhov**

B.I. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., 61103 Kharkov, Ukraine

E-mail: kuplevakhsky@ilt.kharkov.ua

*Received April 30, 2010*

**Аннотация**

We present the exact and complete analytical solution of the classical problem of the Josephson tunnel junction
of arbitrary length *W* Î (0, ∞) in the presence of externally applied magnetic fields and transport currents.
Contrary to a wide-spread belief, the exact analytical solution unambiguously proves the absence of any *qualitative* difference between the 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 junction's edges. We also refine on the mathematical definition of critical transport current.

PACS: 74.50.+r Tunneling phenomena; Josephson effects;

PACS: 03.75.Lm Tunneling, Josephson effect, Bose–Einstein condensates in periodic potentials, solitons, vortices,and topological excitations;

PACS: 02.30.Oz Bifurcation theory.

**Ключевые слова: **Josephson tunnel junction, vortex structure, critical current.