  We develop a theory of optical frequency comb generation in ultra-compact
Surface Nanoscale Axial Photonic (SNAP) bottle microresonators, employing the
nonlinear interaction of whispering gallery modes which are confined along an
optical fiber with nanoscale radius variation. We predict that a SNAP
microresonator with a few-$\mu$m radius can generate a frequency comb with an
ultra-fine sub-GHz spectral spacing, which would require traditional ring
resonators of $cm$ radius. We identify regimes of stable or quasi-periodic comb
dynamics due to soliton excitation, and also show that special engineering of
the SNAP radius profile can be used to compensate nonlinearity induced
dispersion.
