5 The unperturbed soliton parameters are [25] where ξ ≡ T$_{s}$/J$_{s}$S is the inverse saturation power. We have simulated Eqs. (1,2,4) in the case of J$_{s}$ =50 µ J/cm 2 and T$_{s}$ =0.5 ps, that correspond to the measure- ment in Fig. 1. Two cases have been considered: weak focusing ( A$_{eff}$ =4000 µ m 2 or saturation energy E$_{s}$ =2 nJ) and hard focusing ( A$_{eff}$ =1000 µ m 2 or saturation energy E$_{s}$ =0.5 nJ). We also considered Eq. (5) for the same peak saturation level as weakly focused SA (i.e. ξ − $^{1}$= 4 kW). In the latter case the SA effectively becomes in- stantaneous, and the perturbed soliton spectrum is the same as for the Kerr-lens modelocking, i.e. as for the cubic-quintic ˆ Σ. When the saturation energy is suffi- ciently large, there is no difference between the models expressed by Eqs. (4) and (5). The effect of a narrow absorption line is similar to that of the soliton of the cubic-quintic Eq. (1). Decrease of the saturation energy E$_{s}$ causes down-shift of the pulse spectrum as a whole, but the narrow bend on the soliton spectrum reproduces the real part of the absorber permittivity. One can thus conclude that the type of the SAM is irrelevant for an ef- fect of the narrowband absorption lines on a dissipative soliton spectrum. Another important conclusion from the numerical sim- ulations is the demonstrated stability of the dissipative soliton against perturbations induced by narrowband ab- sorption. In the following analytical treatment we shall therefore omit the stability analysis. To study the transformation of dissipative soliton spec- trum under action of narrow absorption lines, we ap- ply the perturbation method [9, 25]. Since the ba- sic features already become apparent for the perturbed Schr¨ odinger soliton and do not depend on SAM details, we shall consider the simplest case of the cubic nonlin- ear gain ˆ Σ[ P ] = κP . The unperturbed solitonic chirp- free solution of such reduced equation with ˆ Γ = 0 is a ( z, t ) = A$_{0}$ sech ( t/T$_{0}$ ) exp [ iφ ( t ) + iqz ] with dφ/dt = ϖ . In the spectral domain, Eq. (9) becomes Linearization of the latter with respect to a perturbation f ( t ) exp ( iqz ) results in Hence, the soliton wavenumber is q = − γσ/κ . Its is reasonable to treat the soliton of the reduced Eq. (1) as the Schr¨ odinger soliton with the parameters con- strained by the dissipative terms σ , α and κ (see Eqs. (6,7)). This implies that the equation, which has to be linearized with respect to a small perturbation co- propagating with the soliton without beating, decay or growth (i.e. having a wavenumber real and equals to q [9]), is the perturbed nonlinear Schr¨ odinger equation where [25] ∂A ( z, t ) ∂z = i β$_{2}$ 2 ∂ 2 ∂t $_{2}$A ( z, t ) − iγP ( z, t ) A ( z, t ) + ˆ Γ [ A ( z, t )] . (8) iqf ( t ) = i β$_{2}$ 2 d $^{2}$f ( t ) dt 2 − iγ [ 2 | a ( t ) | 2 f ( t ) + a ( t ) 2 f $^{∗}$( t ) ] + ˆ Γ ( a + f ) . (9) [ k ( ω ) − q ] ˜ f ( ω ) + 1 π ∞ ∫ −∞ dω $^{′}$U ( ω − ω ′ ) ˜ f ( ω ′ ) + 1 2 π ∞ ∫ −∞ dω $^{′}$U ( ω − ω ′ ) ˜ f ∗ ( ω ′ ) = S ( ω ) , (10) U ( ω ) = − πγT 2 $_{0}$A 2 $_{0}$ω csch ( πT$_{0}$ω/ 2) , S ( ω ) = iA$_{0}$πT$_{0}$ cosh ( πT$_{0}$ω/ 2) ∑ l ϵ$_{l}$ 1 − i ( ω − ω$_{l}$ ) / Ω$_{l}$ 1 + ( ω ω$_{l}$ ) 2 / Ω $_{2}$, (11) k ( ω ) = − β$_{2}$ 2 ω 2 − ∑ l ϵ$_{l}$ ( ω − ω$_{l}$ ) / Ω$_{l}$ + i 1 + ( ω − ω$_{l}$ ) 2 / Ω $_{2}$. ϖ = 0 , A 2 0 = 2 α κT 2 0 , (6) q = β$_{2}$ 2 T 2 0 , T$_{0}$ = √ α σ , β$_{2}$ = − 2 αγ κ , (7) σ > 0 . where the equation parameters are confined 4. PERTURBATIVE ANALYSIS OF SOLITON SPECTRUM