  The problem of vortex structure in a single Josephson junction in an external
magnetic field, in the absence of transport currents, is reconsidered from a
new mathematical point of view. In particular, we derive a complete set of
exact analytical solutions representing all the stationary points (minima and
saddle-points) of the relevant Gibbs free-energy functional. The type of these
solutions is determined by explicit evaluation of the second variation of the
Gibbs free-energy functional. The stable (physical) solutions minimizing the
Gibbs free-energy functional form an infinite set and are labelled by a
topological number Nv=0,1,2,... Mathematically, they can be interpreted as
nontrivial ''vacuum'' (Nv=0) and static topological solitons (Nv=1,2,...) of
the sine-Gordon equation for the phase difference in a finite spatial interval:
solutions of this kind were not considered in previous literature. Physically,
they represent the Meissner state (Nv=0) and Josephson vortices (Nv=1,2,...).
Major properties of the new physical solutions are thoroughly discussed. An
exact, closed-form analytical expression for the Gibbs free energy is derived
and analyzed numerically. Unstable (saddle-point) solutions are also classified
and discussed.

Superconductivity;Pattern Formation and Solitons