Josephson Vortices in Weakly-Coupled Superconductors as Topological Solitons of the Sine-Gordon Equations for Phase Differences

Abstract

We present a mathematically rigorous solution of the problem of magnetic properties of weakly-coupled superconducting multilayers with an arbitrary number N ≥ 2 of superconducting layers in external parallel magnetic fields H > 0. By minimizing a relevant Gibbs free-energy functional, we show that the equilibrium vortex structure is given by a new class of soliton solutions: namely, topological solitons of a system of N − 1 coupled static sine-Gordon equations for the phase differences in a finite spatial interval I = [−L,L]. A complete classification of the new soliton solutions is presented. For N = 2, 3, and N = ∞, exact, closed-form analytical expressions are derived. The special case L = ∞, H = 0 is considered separately. Non-soliton solutions are also analyzed: they are shown to be saddle points of the Gibbs free-energy functional. A comparison with the experiment is drawn.