  Frequency-modulated (FM) combs with a linearly-chirped frequency and nearly
constant intensity occur naturally in certain laser systems; they can be most
succinctly described by a nonlinear Schr\"odinger equation with a phase
potential. In this work, we perform a comprehensive analytical study of FM
combs in order to calculate their salient properties. We develop a general
procedure that allows mean-field theories to be constructed for arbitrary sets
of master equations, and as an example consider the case of reflective defects.
We derive an expression for the FM chirp of arbitrary Fabry-Perot
cavities$-$important for most realistic lasers$-$and use perturbation theory to
show how they are affected by gain curvature. Lastly, we show that an
eigenvalue formulation of the laser's dynamics can be useful for characterizing
all of the stable states of the laser: the fundamental comb, the
continuous-wave solution, and the harmonic states.
