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tive chirp. In this study, we consider only the negative dispersion regime, the positive dispersion regime will be a subject of following studies. 

The results of the simulation are shown in Figs. 3 and 4, and they demonstrate the same dispersion-like modu- lation of the pulse spectrum. Fig. 3 demonstrates action by three narrow (Ω =2 GHz) absorption lines centered at -10, 0 and 10 GHz in the neighborhood of ω = 0. One can see (Fig. 3a), that the absorption lines do not cause spectral dips at ω$_{l}$ , but produce sharp bends, very much like the case of the true perturbed Schr¨ odinger soli- ton considered before. One can also clearly see the col- lective redistribution of spectral power from higher- to lower-frequencies, which enhances local spectral asym- metry (Fig. 3a). Such an asymmetry suggests that the dominating contribution to a soliton perturbation results from the real part of an absorber permittivity, which, in particular, causes the time-asymmetry of perturbation in the time domain. This asymmetry is seen in time domain as a ns-long modulated exponential precursor in Fig. 3b. 

The simulated effect of a single narrow absorption line centered at ω = 0 is shown in Fig. 4 for different values of peak absorption ϵ and width Ω. In Fig. 4a, ϵ = − 0 . 05 and Ω = 4 GHz (solid curve, open circles and crosses) and 1 GHz (dashed curve, open squares and triangles). The solid and dashed curves demonstrate the action of complex profile (2), whereas circles (squares) and crosses (triangles) demonstrate the separate action of ℑ ( ˆ Γ) and ℜ ( ˆ Γ), respectively. One can see, that the profile of per- turbed spectrum traces that formed by only ℑ ( ˆ Γ) (i.e. it traces the real part of an absorber permittivity). One can say, that the pure phase effect ( ℑ ( ˆ Γ), circles and squares in Figs. 4a,b) strongly dominates over the pure absorption ( ℜ ( ˆ Γ), crosses in Figs. 4a,b), like that for the Schr¨ odinger soliton. Such a domination enhances with a 

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lowered ϵ (Fig. 4b, crosses), however the | ϵ | growth in- creases the relative contribution of ℜ ( ˆ Γ) and causes the frequency downshift of the bend (Fig. 4a). Amplitude of the bend traces the ϵ value, while its width is defined by Ω. 

The SAM considered above is modeled by the cubic- quintic nonlinear term ˆ Σ in Eq.(1). Such a SAM is typ- ically realized by using the self-focusing inside an ac- tive medium (Kerr-lens modelocking). For reliable self- starting operation of mode-locked oscillator it is often de- sirable to use a suitable saturable absorber (SA), e.g. a semiconductor-based SESAM [23, 24]. Such an absorber can be described in the simplest case by a single-lifetime two-level model, giving the time-dependent loss coeffi- cient Λ ( t ) as 

where Λ$_{0}$ is the loss coefficient for a small signal, A$_{eff}$ is the effective beam area on the SA, T$_{s}$ and J$_{s}$ are the SA relaxation time and the saturation energy fluency, respectively. Eq. (4) supplements Eq. (1) and the SAM term ˆ Σ in the latter has to be replaced by (Λ$_{0}$ − Λ ( t )). When the pulse width is longer than the SA relaxation time, one can replace Eq. (4) by its adiabatic solution so that 

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