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 $$\lambda_J$$. 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:

$ \lambda_J^2\phi_{xx}-\omega_p^{-2}\phi_{tt}-\sin(\phi) =\omega_c^{-1}\phi_t - j/j_c, $ 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: $ \phi_{xx}-\phi_{tt}-\sin(\phi)=\alpha\phi_t - \gamma, $ where spatial coordinate is normalized to the Josephson penetration depth $$\lambda_J$$ and time is normalized to the inverse plasma frequency $$\omega_p^{-1}$$. The parameter $$\alpha=1/\sqrt{\beta_c}$$ is the dimensionless damping parameter ($$\beta_c$$ is McCumber-Stewart parameter), and, finally, $$\gamma=j/j_c$$ is a normalized bias current.

Important solutions
$ \phi(x,t)=4\arctan\exp\left(\pm\frac{x-ut}{\sqrt{1-u^2}}\right) $|undefined 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 $$c_0=\lambda_J\omega_p$$, which represent a typical unit of velocity and equal to the unit of space $$\lambda_J$$ divided by unit of time $$\omega_p^{-1}$$.
 * Small amplitude plasma waves. $$\phi(x,t)=A\exp[i(kx-\omega t)]$$
 * Soliton (aka fluxon, Josephson vortex):