Long Josephson junction

In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth formula_1. This definition is not strict.
In terms of underlying model a "short Josephson junction" is characterized by the Josephson phase formula_2, which is only a function of time, but not of coordinates i.e. the Josephson junction is assumed to be point-like in space. In contrast, in a long Josephson junction the Josephson phase can be a function of one or two spatial coordinates, i.e., formula_3 or formula_4.
Simple model: the sine-Gordon equation.
The simplest and the most frequently used model which describes the dynamics of the Josephson phase formula_5 in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:
where subscripts formula_6 and formula_7 denote partial derivatives with respect to formula_6 and formula_7, formula_1 is the Josephson penetration depth, formula_11 is the Josephson plasma frequency, formula_12 is the so-called characteristic frequency and formula_13 is the bias current density formula_14 normalized to the critical current density formula_15. In the above equation, the r.h.s. is considered as perturbation.
Usually for theoretical studies one uses normalized sine-Gordon equation:
where spatial coordinate is normalized to the Josephson penetration depth formula_1 and time is normalized to the inverse plasma frequency formula_17. The parameter formula_18 is the dimensionless damping parameter (formula_19 is McCumber-Stewart parameter), and, finally, formula_20 is a normalized bias current.
Important solutions.
\right)
Here formula_6, formula_7 and formula_24 are the normalized coordinate, normalized time and normalized velocity. The physical velocity formula_25 is normalized to the so-called Swihart velocity formula_26, which represent a typical unit of velocity and equal to the unit of space formula_1 divided by unit of time formula_17.