  The complex Ginzburg-Landau equation has been used extensively to describe
various non-equilibrium phenomena. In the context of lasers, it models the
dynamics of a pulse by averaging over the effects that take place inside the
cavity. Ti:sapphire femtosecond lasers, however, produce pulses that undergo
significant changes in different parts of the cavity during each round-trip.
The dynamics of such pulses is therefore not adequately described by an average
model that does not take such changes into account. The purpose of this work is
severalfold. First we introduce the dispersion-managed Ginzburg-Landau equation
(DMGLE) as an average model that describes the long-term dynamics of systems
characterized by rapid variations of dispersion, nonlinearity and gain in a
general setting, and we study the properties of the equation. We then explain
how in particular the DMGLE arises for Ti:sapphire femtosecond lasers and we
characterize its solutions. In particular, we show that, for moderate values of
the gain/loss parameters, the solutions of the DMGLE are well approximated by
those of the dispersion-managed nonlinear Schrodinger equation (DMNLSE), and
the main effect of gain and loss dynamics is simply to select one among the
one-parameter family of solutions of the DMNLSE.
