In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict. In terms of underlying model a short Josephson junction is characterized by the Josephson phase \phi(t), 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., \phi(x,t) or \phi(x,y,t).

Simple model: the sine-Gordon equation

The simplest and the most frequently used model which describes the dynamics of the Josephson phase \phi in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like: where subscripts x and t denote partial derivatives with respect to x and t, \lambda_J is the Josephson penetration depth, \omega_p is the Josephson plasma frequency, \omega_c is the so-called characteristic frequency and j/j_c is the bias current density j normalized to the critical current density j_c. 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 \lambda_J and time is normalized to the inverse plasma frequency. The parameter is the dimensionless damping parameter (\beta_c is McCumber-Stewart parameter), and, finally, is a normalized bias current.

Important solutions

Here x, t and u=v/c_0 are the normalized coordinate, normalized time and normalized velocity. The physical velocity v is normalized to the so-called Swihart velocity, which represent a typical unit of velocity and equal to the unit of space \lambda_J divided by unit of time.