Josephson vortices and the Meissner effect in stacked junctions and   layered superconductors: Exact analytical results
  We present an exact mathematical description of Josephson vortices and of the Meissner effect in periodic thin-layer superconductor/insulator structures with an arbitrary number of identical junctions N-1 (N is the number of superconducting layers) in terms of localized solutions to a system of differential equations for phase differences. We establish a general criterion of the existence of localized solutions. We show that Meissner solutions are characterized by several Josephson lengths [N/2 lengths for even N, and (N-1)/2 lengths for odd N]. We derive an exact expression for the superheating field of the Meissner state as an explicit function of N. For Josephson vortices, we find two basically different types of topological solutions: ''vortex-plane'' solutions and incoherent vortex solutions. Thermodynamically stable ''vortex-plane'' solutions represent a chain of N-1 vortices (one vortex per each insulating layer). They are characterized by the same set of Josephson lengths as the Meissner solutions. We obtain exact analytical expressions for their self-energy and for the lower critical field. Incoherent vortex solutions comprise solutions with k < N-1 vortices and different vortex-antivortex configurations. In contrast to the ''vortex-plane'' solutions, they prove to be thermodynamically unstable, and their spatial dependence is characterized, in general, by N-1 length scales. As an illustration, we analyze 1-4-Josephson-junction stacks and investigate a transition to the layered superconductor limit. 
Sergey V. Kuplevakhsky (Kharkov National University, Kharkov, Ukraine)