A quantum flux, a 2 pi fluxon, described by a sine-Gordon equation is an el
ementary excitation in long Josephson junctions. We investigated the fluxon
dynamics in isolated long Josephson junctions. In such isolated systems, t
he total number of Cooper pairs is conserved, so the fluxon dynamics map ch
ange. We found that the fluxons obey a double sine-Gordon (DSG) equation ba
sed on this change in the Josephson acceleration relation, i.e., the voltag
e-phase difference relation. It should be possible to confirm the existence
of DSG fluxons by detecting the relative oscillations of two pi fluxons. (
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