Time dilation of a bound half-fluxon pair in a long Josephson junction with a ferromagnetic insulator

The fluxon dynamics in a long Josephson junction with a ferromagnetic insulating layer is investigated. It is found that the Josephson phase obeys a double sine-Gordon equation involving a bound pi fluxon solution, and the internal oscillations of the bound pair acting as a clock exhibit Lorentz reductions in their frequencies regarded as a relativistic effect in the time domain, i.e., time dilation. This is the complement to the Lorentz contraction of fluxons with no clock. A possible observation scheme is also discussed.