Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2008-10-14
Nonlinear Sciences
Pattern Formation and Solitons
22 pages, 4 figures, to appear in Nonlinearity
Scientific paper
10.1088/0951-7715/21/12/007
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.
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