A study is made of frequency comb generation described by the driven and
damped nonlinear Schr\"odinger equation on a finite interval. It is shown that
frequency comb generation can be interpreted as a modulational instability of
the continuous wave pump mode, and a linear stability analysis, taking into
account the cavity boundary conditions, is performed. Further, a truncated
three-wave model is derived, which allows one to gain additional insight into
the dynamical behaviour of the comb generation. This formalism describes the
pump mode and the most unstable sideband and is found to connect the coupled
mode theory with the conventional theory of modulational instability. An
in-depth analysis is done of the nonlinear three-wave model. It is demonstrated
that stable frequency comb states can be interpreted as attractive fixed points
of a dynamical system. The possibility of soft and hard excitation states in
both the normal and the anomalous dispersion regime is discussed.
Investigations are made of bistable comb states, and the dependence of the
final state on the way the comb has been generated. The analytical predictions
are verified by means of direct comparison with numerical simulations of the
full equation and the agreement is discussed.Comment: 9 pages, 6 figures, submitted to Phys. Rev.