  We theoretically investigate the dynamics and stability of a temporal cavity
soliton (CS) excited inside a silicon-based microresonator that exhibits
free-carrier generation as a result of two-photon absorption (TPA). The optical
propagation of the CS is modeled through a mean-field Lugiato-Lefever equation
(LLE) coupled with an ordinary differential equation accounting for the
generation of free carriers owing to TPA. The CS experiences several
perturbations (like intrapulse Raman scattering (IRS), TPA, free-carrier
absorption (FCA), free-carrier dispersion (FCD), etc.) during its round-trip
evolution inside the cavity. We develop a full variational analysis based on a
Ritz optimization principle which is useful in deriving simple analytical
expressions describing the dynamics of individual pulse parameters of the CS
under perturbation. TPA and FCA limit the efficient comb generation and modify
the stability condition of the CS. We determine the critical condition of
stability modified due to TPA and derive closed-form expressions of the
saturated amplitude and width of stable CS. We perform detailed
modulation-instability analysis and obtain stability condition against
perturbations of steady-state solution of LLE. The CS experiences FCD which
leads to a temporal acceleration resulting in spectral blueshift. Exploiting
the variational analysis, we estimate these temporal and spectral shifts. We
also include IRS in our perturbation theory and analytically estimate the
frequency redshifting. Finally, we study the effect of pump-phase-modulation on
a stable CS. All our analytical results are found to be in good agreement with
the data obtained from the full numerical solution of LLE.
