# Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions

## 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 labeled by a topological number N{sub v}=0,1,2,... . Mathematically, they can be interpreted as nontrivial 'vacuum' (N{sub v}=0) and static topological solitons (N{sub v}=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 (N{sub v}=0) and Josephson vortices (N{sub v}=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.

- Authors:

- B. I. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Avenue, 61103 Kharkov (Ukraine)

- Publication Date:

- OSTI Identifier:
- 20787833

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. B, Condensed Matter and Materials Physics; Journal Volume: 73; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevB.73.024513; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 36 MATERIALS SCIENCE; ANALYTICAL SOLUTION; ELECTRIC CURRENTS; EQUILIBRIUM; FREE ENERGY; FREE ENTHALPY; JOSEPHSON EFFECT; JOSEPHSON JUNCTIONS; MAGNETIC FIELDS; SINE-GORDON EQUATION; SOLITONS; TOPOLOGY; VORTICES

### Citation Formats

```
Kuplevakhsky, S. V., and Glukhov, A. M.
```*Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVB.73.0.

```
Kuplevakhsky, S. V., & Glukhov, A. M.
```*Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions*. United States. doi:10.1103/PHYSREVB.73.0.

```
Kuplevakhsky, S. V., and Glukhov, A. M. Sun .
"Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions". United States.
doi:10.1103/PHYSREVB.73.0.
```

```
@article{osti_20787833,
```

title = {Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions},

author = {Kuplevakhsky, S. V. and Glukhov, A. M.},

abstractNote = {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 labeled by a topological number N{sub v}=0,1,2,... . Mathematically, they can be interpreted as nontrivial 'vacuum' (N{sub v}=0) and static topological solitons (N{sub v}=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 (N{sub v}=0) and Josephson vortices (N{sub v}=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.},

doi = {10.1103/PHYSREVB.73.0},

journal = {Physical Review. B, Condensed Matter and Materials Physics},

number = 2,

volume = 73,

place = {United States},

year = {Sun Jan 01 00:00:00 EST 2006},

month = {Sun Jan 01 00:00:00 EST 2006}

}