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The problem of vortex structure in a single Josephson junction in an externalmagnetic field, in the absence of transport currents, is reconsidered from anew mathematical point of view. In particular, we derive a complete set ofexact analytical solutions representing all the stationary points (minima andsaddle-points) of the relevant Gibbs free-energy functional. The type of thesesolutions is determined by explicit evaluation of the second variation of theGibbs free-energy functional. The stable (physical) solutions minimizing theGibbs free-energy functional form an infinite set and are labelled by atopological number Nv=0,1,2,... Mathematically, they can be interpreted asnontrivial ''vacuum'' (Nv=0) and static topological solitons (Nv=1,2,...) ofthe 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. Anexact, closed-form analytical expression for the Gibbs free energy is derivedand analyzed numerically. Unstable (saddle-point) solutions are also classifiedand discussed.Comment: 17 pages, 4 Postscript figure

Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions