In the previous chapter we have seen that both constant dispersion and dispersion-managed solitons are important pulses in the study of long-distance transmission/communications. It is noteworthy that experiments reveal that solitons or localized pulses are also present in mode-locked (ML) lasers. Femtosecond solid-state lasers, such as those based on the Ti:sapphire (Ti:S) gain medium, and fiber ring lasers have received considerable attention in the field of ultra-fast science. In the past decade, following the discovery of mode-locking, the improved performance of these lasers has led to their widespread use, cf. Cundiff et al. (2008). In most cases interest in ultra-short pulse mode-locking has been in the net anomalous dispersive regime. But mode-locking has also been demonstrated in fiber lasers operating in the normal regime. Mode-locking operation has been achieved with relatively large pulse energies (Ilday et al., 2004b,a; Chong et al., 2008b).
In our investigations we have employed a distributive model, termed the power-energy saturation (PES) equation (cf. Ablowitz et al., 2008; Ablowitz and Horikis, 2008, 2009b; Ablowitz et al., 2009c). This model goes beyond the well-known master laser equation, cf. Haus (1975, 2000), in that it contains saturable power (intensity) terms; i.e., terms that saturate due to large field amplitudes. This equation has localized pulses that propagate and mode-lock in both an anomalous and a normally dispersive laser for both in the constant as well as dispersion-managed system. This is consistent with recent experimental observations (Ilday et al., 2004b; Chong et al., 2008a).