  We present an exact mathematical description of the Meissner effect and of
the vortex state in periodic thin-layer superconductor/insulator structures
with an arbitrary number of identical junctions N-1 (N is the number of
superconducting layers) in the presence of a static parallel external field H.
Based on an analytical analysis of the coupled static sine-Gordon (SG)
equations for the phase differences, we obtain a complete classification of all
possible types of physical solutions. We prove that at H>0 these equations
admit only Meissner solutions and topological ''vortex-plane'' solutions. Both
the types of solutions are characterized by [N/2] Josephson lengths. We derive
an explicit analytical expression for the superheating field of the Meissner
state, Hs, as a function of N and show that Hs simultaneously determines the
penetration field for the vortex planes. Thermodynamically stable
''vortex-plane'' solutions represent coherent chains of Josephson vortices (one
vortex per each insulating layer in a chain). Being a natural generalization of
ordinary Josephson vortices in a single junction, the vortex-plane solutions
inherit such properties of the former as periodicity along the layers and the
overlapping of states with different topological numbers. We obtain exact
analytical expressions for the self-energy of a vortex plane and for the lower
critical field Hc1. In contrast to a prevailing view, the coupled SG equations
do not possess any single-vortex solutions for H>0. The general consideration
is illustrated by two exactly-solvable examples (N=2,3). Experimental
implications of the results are discussed.
