Static Solitons of the Sine-Gordon Equation and Equilibrium Vortex   Structure in Josephson Junctions
  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. 
S. V. Kuplevakhsky and A. M. Glukhov (B.Verkin Institute for Low   Temperature Physics and Engineering, Kharkov, Ukraine)