2 2. THE MODEL Our approach is based on the treatment of an ultra- short pulse as one-dimensional dissipative soliton of the nonlinear complex Ginzburg-Landau equation (CGLE) [8, 9]. This equation has such a wide horizon of appli- cation that the concept of “the world of the Ginzburg- Landau equation” has become broadly established [10]. In particular, such a model describes a pulse with the duration T$_{0}$ inside an oscillator or propagating along a nonlinear fiber. To obey the CGLE, the electromagnetic field with the amplitude A ( z, t ) should satisfy the slowly-varying ampli- tude approximation, provided by the relation ω$_{0}$ ≫ 1 /T$_{0}$ , where ω$_{0}$ is the field carrier frequency, t is the local time, and z is the propagation coordinate. This approximation is well satisfied even for pulses of nearly single optical cy- cle duration [11]. When additionally we can neglect the field variation along the cavity round-trip or the varia- tion of material parameters along a fiber, as well as the contribution of higher-order dispersions, the amplitude dynamics can be described on the basis of the general- ized CGLE [9, 12–14] where P ≡ | A | 2 is the instant field power and α is the inverse gain bandwidth squared. The nonlinear terms in Eq. (1) describe i) saturable self-amplitude modu- lation (SAM) with nonlinear gain defined by the non- linear operator ˆ Σ, and ii) self-phase modulation (SPM), defined by the parameter γ . For a laser oscillator, γ = 4 πnn$_{2}$l$_{cryst}$/λ$_{0}$A$_{eff}$ . Here λ$_{0}$ is the wavelength; n and n$_{2}$ are the linear and nonlinear refractive indexes of an active medium, respectively; l$_{cryst}$ is the length of the ac- tive medium; A$_{eff}$ = πw 2 is the effective area of a Gaus- sian mode with the radius w inside the active medium. The propagation coordinate z is naturally normalized to the cavity length, i.e. z becomes the cavity round-trip number. For a fiber propagation, γ = 2 πnn$_{2}$/λ$_{0}$A$_{eff}$ , where n and n$_{2}$ are the linear and nonlinear refractive indexes of a fiber, respectively, and A$_{eff}$ is the effective mode area of the fiber [15]. Finally, β$_{2}$ is the round-trip net group delay dispersion (GDD) for an oscillator or the group velocity dispersion parameter for a fiber with β$_{2}$ < 0 corresponding to anomalous dispersion. The typical explicit expressions for ˆ Σ[ P ] in the case, when the SAM response is instantaneous, are i) ˆ Σ[ P ] = κP (cubic nonlinear gain), ii) ˆ Σ[ P ] = κ ( P − ζP $^{2}$) (cubic-quintic nonlinear gain), and iii) ˆ Σ[ P ] = κP/ (1 + ζP ) (perfectly saturable nonlinear gain) [9, 16]. The second case corresponds to an oscillator mode-locked by the Kerr-lensing [8]. The third case rep- resents, for instance, a response of a semiconductor sat- urable absorber, when T$_{0}$ exceeds its excitation relax- ation time [17]. However, if the latter condition is not satisfied, one has to add an ordinary differential equation for the SAM and Eq. (1) becomes an integro-differential equation (see below). The σ -term is the saturated net-loss at the carrier frequency ω$_{0}$ , which is the reference frequency in the model. This term is energy-dependent: the pulse energy E ( z ) ≡ ∫ ∞ −∞ P ( z, t ) dt can be expanded in the vicinity of threshold value σ = 0 as σ ≈ δ ( E/E ∗ − 1) [18], where δ = ℓ $^{2}$/g$_{0}$ ( ℓ is the frequency-independent loss and g$_{0}$ is the small-signal gain, both for the round-trip) and E ∗ is the round-trip continuous-wave energy equal to the av- erage power multiplied by the cavity period. The operator ˆ Γ describes an effect of the frequency- dependent losses, which can be attributed to an absorp- tion within the dissipative soliton spectrum. That can be caused, for instance, by the gases filling an oscilla- tor cavity or the fiber impurities for a fiber oscillator. Within the framework of this study, we neglect the ef- fects of loss saturation and let ˆ Γ be linear with respect to A ( z, t ). The expression for ˆ Γ [ A ( z, t )] is more conve- nient to describe in the Fourier domain, ˜ A ( z, ω ) being the Fourier image of A ( z, t ). If the losses result from the l independent homogeneously broadened lines centered at ω$_{l}$ (relative to ω$_{0}$ ) with linewidths Ω$_{l}$ and absorption coefficients ϵ$_{l}$ < 0, then the action of operator ˆ Γ can be written down in the form of a superposition of causal Lorentz profiles [19, 20] In the more general case the causal Voigt profile has to be used for ˆ Γ [ ˜ A ] [21]. Causality of the complex profile of Eq. (2) demonstrates itself in the time domain, where one has The conventional analysis of perturbed soliton prop- agation includes approximation of the effective group ∂A ( z, t ) ∂z = { − σ + ˆ Σ[ P ( z, t )] − iγP ( z, t ) } A ( z, t ) + ( α + i β$_{2}$ 2 ) ∂ 2 ∂t $_{2}$A ( z, t ) + ˆ Γ [ A ( z, t )] , (1) ˆ Γ [ ˜ A ] = ( ∑ l ϵ$_{l}$ 1 + i ( ω − ω$_{l}$ ) / Ω$_{l}$ ) ˜ A ( z, ω ) . (2) ˆ Γ [ A ( z, t )] ∝ ∑ l ϵ$_{l}$ Ω$_{l}$ t ∫ −∞ e − (Ω l − iω l ) ( t − t ′ ) A ( z, t $^{′}$) dt $^{′}$. (3)