Static Solitons of the Sine-Gordon Equation and Equilibrium Vortex Structure in Josephson Junctions

TL;DR

This paper provides a comprehensive set of exact analytical solutions for vortex structures in Josephson junctions, interpreting them as topological solitons of the sine-Gordon equation, and analyzes their stability and physical significance.

Contribution

It introduces a complete set of exact solutions for vortex structures in Josephson junctions, including previously unconsidered topological solitons, and analyzes their stability and energy properties.

Findings

Infinite set of stable solutions labeled by topological number Nv

Exact analytical expressions for Gibbs free energy derived

Classification of stable and unstable solutions provided

Abstract

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.