Shape waves
Encyclopedia
Shape waves are excitations propagating along Josephson vortices or fluxons. In the case of two-dimensional Josephson junctions (thick long Josephson junction
Long Josephson junction
In superconductivity, a long Josephson junction is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict....

s with an extra dimension) described by the 2D sine-Gordon equation
Sine-Gordon equation
The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...

, shape waves are distortions of a Josephson vortex
Josephson vortex
In superconductivity, a Josephson vortex is a vortex of supercurrent in a long Josephson junction . The supercurrents circulate around the vortex center which is situated inside the Josephson barrier...

 line of an arbitrary profile. Shape waves have remarkable properties exhibiting Lorentz contraction and time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

 similar to that in special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. Position of the shape wave excitation on a Josephson vortex
Josephson vortex
In superconductivity, a Josephson vortex is a vortex of supercurrent in a long Josephson junction . The supercurrents circulate around the vortex center which is situated inside the Josephson barrier...

acts like a “minute-hand” showing the time in the rest-frame associated with the vortex. At some conditions, a moving vortex with the shape excitation can have less energy than the same vortex without it.
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