Voltage-less alternating current (AC) Josephson effect in two-band superconductors

The sliding motions of inter-band phase difference solitons (i-solitons) cause time-dependent superconducting phase variations in two-band superconductors. Two different voltages cannot exist at the same spatial location, although the conventional AC Josephson effect demands the presence of a voltage difference between the two bands at the same location, if the time-variation of the phase generates the voltage, for which a physical interpretation is not possible. Instead of the conventional AC Josephson effect, a non-linear Klein-Gordon equation (sine-Gordon equation) can enable the dissipationless movement of i-solitons, without voltage generation.