  We apply the moment method to nonlinear pulse propagation in silicon
waveguides in the presence of two-photon absorption, free-carrier dispersion
and free-carrier absorption. The evolution equations for pulse energy, temporal
position, duration, frequency shift and chirp are obtained. We derive analytic
expressions for the free-carrier induced blueshift and acceleration and show
that they depend only on the pulse peak power. Importantly, these effects are
independent of the temporal duration. The moment equations are then numerically
solved to provide fast estimates of pulse evolution trends in silicon photonics
waveguides. We find that group-velocity and free-carrier dispersion dominate
the pulse dynamics in photonic crystal waveguides. In contrast, two-photon and
free-carrier absorption dominate the temporal dynamics in silicon nanowires. To
our knowledge, this is the first time the moment method is used to provide a
concise picture of multiphoton and free-carrier effects in silicon photonics.
The treatment and conclusions apply to any semiconductor waveguide dominated by
two-photon absorption.
