  We investigated quantized modes of kinks in the phase space of
superconducting gaps in a superconductor with multiple gaps. The kink is
described by the sine-Gordon model in a two-gap superconductor and by the
double sine-Gordon model in a three-gap superconductor. A fractional-flux
vortex exists at the edge of the kink, and a fractional-flux vortex will be
stable in a three-gap superconductor with time-reversal symmetry breaking. The
kink and fractional-flux vortex exhibit massless modes as a sliding motion. We
show further that there are one zero-energy mode (massless mode) and quantized
excitation modes in kinks, which are characteristic features of multi-gap
superconductors. The equation of quantized modes for the double sine-Gordon
model is solved numerically. The correction to the ground-state energy is
calculated based on the renormalization theory.
