1 Introduction

Mode-locked fiber lasers with saturable absorbers have attracted a great deal of attention in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13,14], because they stand for ideal sources of high-intensity short pulses used in a vast area of modern communication technology. In these devices, an intensity fluctuation acts in conjunction with nonlinearity of the optical amplifier to modulate the cavity loss, without need for some external control [8, 9, 11,12,13]. Though several mode-locking techniques are reported in the literature, for applications requiring ultrashort pulses produced at finite repetition rates passive mode-locking remains the most preferred technique [1,2,3, 13]. Passive mode-locking rests mainly on an appropriate choice of the optical gain medium, to this last point passively mode-locked fiber lasers with rare-earth-doped optical amplifiers have demonstrated high efficiency in femtosecond pulse generation. In particular, they play a relevant role in pulse multiplexing and applications involving soliton-train structures [15,16,17,18], owing to their optical nonlinearity which can be scaled up (i.e., increased) or in general controlled by doping with rare-earth ions such as erbium [19]. Passive mode-locking has equally been achieved in monolithic semiconductor lasers [20] and in some external cavity semi-conductor lasers [21,22,23], which in general are either in tapered lasers or ring lasers configurations. Using mode-locking techniques these lasers have been found to generate pulse frequencies in the order of tens to hundreds of Gigahertz with ultrashort pulse rates in the order of picoseconds, and femtoseconds.

The self-starting dynamics of passively mode-locked lasers with saturable absorber has been discussed in several works [1,2,3, 10, 13]. Pioneering theoretical attempt on this issue, Haus suggested [1] two possible pictures for self-starting. The first involves direct simulations of the entire evolution of the optical light starting from noise, and the second picture focuses on the evolution of light intensity from continuous wave (cw). This second picture is mainly analytical, and rests on the assumption of a master equation describing the propagation of the optical field over laser roundtrips. Note that if numerical simulations provide details about the evolution of the optical field from noise to either pulse or chaotic regimes [24, 25], they do not offer a global picture of the field evolution over a relatively broad range of values of characteristic parameters of the system. On the contrary, being analytical, the second picture lays emphasis on the spatio-temporal modulation of an input field of arbitrary intensity, over the cavity roundtrips. In particular, it has the merit to involve cw as one of it operation regimes, the instability of which eventually leads to the pulse regime consistently with the spirit of the theory of modulational instability [11].

In the most general context when the gain is inhomogeneous, the dynamics of passively mode-locked fiber lasers with saturable absorber can be described by a complex Ginzburg–Landau equation (CGLE) coupled to a two-level type equation [1,2,3, 8,9,10,11,12,13,14]. In Ref. [11] one such model was considered for passively mode-locked fiber lasers with a Kerr nonlinear gain medium. By following the evolution of an input field from cw to pulse regimes within the framework of the modulational-instability theory, the authors [11] proposed an effective simple approach to self-starting consisting of a global picture in which the real and imaginary parts of the modulation gain, over a broad range of values of the modulation frequency, are mapped onto a two-dimensional complex parameter space. From this map it emerged that increasing characteristic parameters such as the relaxation time, the bandwidth, the group-velocity dispersion coefficient, the saturable absorption coefficient, the Kerr nonlinearity coefficient and the small-signal gain, favor laser self-starting for an input field above a critical threshold power.

However, as stressed above, in fiber lasers the strength of nonlinearity of the optical amplifier can be controlled by doping with erbium ions. This is to say the study carried out in Ref. [11] is valid only for fiber lasers with relatively weak nonlinear optical amplifiers, so weak that the laser propagation can readily be approximated by a CGLE with cubic nonlinearity. Optical media with stronger nonlinearities, including cubic-quintic nonlinear optical media or in general transparent media with saturable optical nonlinearities, possess distinct features as for instance the relevant fact that they are prone to multi-periodic pulses and bound soliton states [13, 26,27,28,29,30,31,32]. Investigating the laser self-starting dynamics in strongly nonlinear passively mode-locked fiber lasers is therefore most desired, for this issue has not been considered so far. Such study is expected to provide a picture of self-starting with some features which are distinct from predictions with models of passively mode-locked fiber lasers characterized by a Kerr nonlinearity.

In this work, we are interested in the self-starting dynamics of a model of passively mode-locked fiber laser with saturable absorber, for which the rate equation is described by a CGLE with a saturable nonlinearity. We shall also pay attention to the system dynamics in the full nonlinear regime, where the laser field is expected to display profiles of high-intensity pulses. Our analysis of laser self-starting will follow the same approach as in Ref. [11], the main objective being to underline the influence of nonlinearity saturation on the self-starting dynamics and specifically on the cw stability regions in the complex parameter space. As for the nonlinear dynamics, numerical simulations will be carried out to solve the complete set of coupled equations describing time evolutions of the laser amplitude and phase, as well as the gain.

2 Cw solution

Consider the self-starting dynamics of a passively mode-locked laser with fast saturable absorber, for which the gain medium is an optical fiber with saturable nonlinearity. The field propagation equation is described by the following CGLE with a saturable nonlinearity:

$$\begin{aligned} \frac{\partial U}{\partial z}= \left( g - \ell + i\theta \right) U + \left( B + iD\right) \frac{\partial ^2 U}{\partial t^2} + \frac{\varGamma +iK}{1+\gamma \vert U\vert ^2}\vert U\vert ^2 U, \end{aligned}$$
(1)

where U(z, t) is the optical field, z is the cavity roundtrip number, t is the laser propagation time, g is the gain, \(\ell\) is the constant loss and \(\theta\) is the phase change over each roundtrip. The characteristic parameters B and D are, respectively, the spectral filter and group-delay dispersion, \(\varGamma\) and K are the fast saturable absorber and nonlinearity coefficients, respectively, and \(\gamma\) accounts for nonlinearity saturation in the fiber amplifier.

The gain dynamics will be described by the two-level equation [11, 33, 34]:

$$\begin{aligned} \frac{\mathrm{d}g}{\mathrm{d}t}=-\frac{(g-g_0)}{T_0}-\frac{|U(z,t)|^2}{T_0 P_\mathrm{s}}g, \end{aligned}$$
(2)

where \(g_0\) is the homogeneous gain, \(P_\mathrm{s}\) is the saturation power and \(T_0\) is the gain relaxation time. Note that when \(\gamma =0\) Eq. (1) reduces to the rate equation considered in Ref. [11]. For small \(\gamma\), the last term in Eq. (1) can be expanded leading among others to the CGLE with a cubic-quintic nonlinearity [35,36,37].

To analyze the laser self-starting features for the present model within the framework of the modulational-instability approach, we first consider its cw regime in the steady state. In this regime, the coupled set Eqs. (1) and (2) admits solutions of the forms:

$$\begin{aligned} U(z)= \sqrt{P_\mathrm{c}}\,\mathrm{e}^{i q_\mathrm{s} z},\quad g(t)= g_\mathrm{s}, \end{aligned}$$
(3)

where \(P_\mathrm{c}=U_\mathrm{c}^2\) in the input power, \(q_\mathrm{s}\) is the cw wavenumber and \(g_\mathrm{s}\) is the steady-state gain. Replacing these in Eqs. (1) and (2) and separating real from imaginary parts, we obtain:

$$\begin{aligned} g_\mathrm{s}= \ell - \frac{\varGamma P_\mathrm{c}}{1+\gamma P_\mathrm{c}} = \frac{g_0}{1 + P_\mathrm{c}/P_\mathrm{s}}, \end{aligned}$$
(4)
$$\begin{aligned} q_\mathrm{s}= \theta + \frac{K P_\mathrm{c}}{1+\gamma P_\mathrm{c}}. \end{aligned}$$
(5)

Equations (4) and (5) determine values of the steady-state gain \(g_\mathrm{s}\) and of the wavenumber \(q_\mathrm{s}\), for which the laser is a plane wave. Equation (4) is particularly relevant given that according to Eq. (1), the stability of cws will depend on the balance between the gain g and the loss \(\ell\). So according to Eq. (4) this balance should be determined by \(P_\mathrm{c}\) and \(\gamma\), for a given saturation power \(P_\mathrm{s}\). Figure 1 summarizes the cw stability in steady state, where the small-signal power gain is defined as \(\text {exp}(2g_0)\) and where we considered three different values of the nonlinearity saturation coefficient \(\gamma\) namely \(\gamma =0\), \(\gamma =0.1\) and \(\gamma =0.5\).

Fig. 1
figure 1

Self-starting region in the \(g_0\)–\(\ell\) plane for \(P_\mathrm{c}=1\), \(P_\mathrm{s}=2\), \(\varGamma =0.01\) and three different values of the nonlinearity saturation coefficient namely \(\gamma =0\), 0.1 and 0.5

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Figure 1 shows a strong broadening of the self-starting region as compared with the case of cubic nonlinearity studied in Ref. [11], when the nonlinearity saturation coefficient \(\gamma\) is increased from zero. It should be noted here that \(\gamma\) is inversely proportional to the cut-off power of the optical field in the gain medium, suggesting that the saturable nonlinearity will be detrimental to cw stability at relatively small input powers.

3 Cw stability

To investigate the stability of the cw Eq. (3), we carry out a modulational-instability analysis by following the evolution of the cw mode bound to a plane-wave noise i.e.:

$$\begin{aligned} U(t)= & {} \left[ \sqrt{P_\mathrm{c}} + \tilde{u}(z,t)\right] \mathrm{e}^{(iq_\mathrm{s} z)}, \nonumber \\ g(t)= & {} g_\mathrm{s} + \tilde{g}. \end{aligned}$$
(6)

Replacing these in Eqs. (1) and (2) and linearizing in u, we obtain:

$$\begin{aligned} \tilde{u}_z= & {} \alpha \tilde{u}_{tt} + \left( \tilde{u} + \tilde{u}^*\right) F_u + \sqrt{P_\mathrm{c}}\tilde{g}, \end{aligned}$$
(7)
$$\begin{aligned} \tilde{g}_t= & {} -\frac{\tilde{g}}{T_\mathrm{e}} + F_g(\tilde{u}, \tilde{u}^*), \end{aligned}$$
(8)

with

$$\begin{aligned} \alpha= & {} B+iD,\quad F_u=P_\mathrm{c}\frac{\varGamma + iK}{(1+\gamma P_\mathrm{c})^2}, \nonumber \\ F_g= & {} -\frac{\epsilon _\mathrm{c}}{T_\mathrm{e}}(\tilde{u}+\tilde{u}^*),\quad \epsilon _\mathrm{c}= \frac{g_0\sqrt{P_\mathrm{c}}}{P_\mathrm{s}(1+P_\mathrm{c}/P_\mathrm{s})^2}, \end{aligned}$$
(9)

and the effective relaxation time \(T_\mathrm{e}\) is defined as:

$$\begin{aligned} T_\mathrm{e}= \frac{T_0}{1+P_\mathrm{c}/P_\mathrm{s}}. \end{aligned}$$
(10)

The inhomogeneous linear first-order ordinary differential equation (8) can be solved by means of the Green function method [38]. This method yields:

$$\begin{aligned} \tilde{g}(t)=\int _{-\infty }^t{G(t,t')F_g[\tilde{u}(t'), \tilde{u}^*(t')]\mathrm{d}t'}, \end{aligned}$$
(11)

where \(G(t,t')= \mathrm{e}^{-(t-t')/T_\mathrm{e}}H(t-t')\) is the Green function, with \(H(t-t')\) the step function. Now if we pick:

$$\begin{aligned} {[}\tilde{u}(z,t), \tilde{u}^*(z,t)]= [A_1, A_2] \mathrm{e}^{(\lambda z + i\omega t)}, \end{aligned}$$
(12)

where \(\lambda\) is the grow rate (or the spatial amplitude modulation) and \(\omega\) is the modulation frequency, Eq. (7) together with its complex conjugate lead to the following secular equation in matrix form:

$$\begin{aligned} \lambda \left( \begin{array}{c} A_1 \\ A_2 \end{array}\right) = \left[ \left( \begin{array}{cc} m_1 &{} m_2\\ m_2^{*} &{} m_1^{*} \end{array} \right) -m_0 \left( \begin{array}{cc} 1&{}1 \\ 1&{}1 \end{array}\right) \right] \; \left( \begin{array}{cc} A_1 \\ A_2 \end{array} \right) , \end{aligned}$$
(13)

with:

$$\begin{aligned} m_1=-\alpha + F_u,\quad m_2= F_u,\quad m_0= \frac{\sqrt{P_\mathrm{c}}\epsilon _\mathrm{c}}{1+i\omega T_\mathrm{e}}. \end{aligned}$$
(14)

The determinant of the above \(2\times 2\) matrix gives rise to a quadratic polynomial in \(\lambda\), the two possible roots of which are given by:

$$\begin{aligned} \lambda _{1,2}= & {} \frac{\varGamma P_\mathrm{c}}{(1+\gamma P_\mathrm{c})^2} - B\omega ^2 - m_0 \nonumber \\&\pm \sqrt{\left[ m_0 - \frac{\varGamma P_\mathrm{c}}{(1+\gamma P_\mathrm{c})^2}\right] ^2 - (D\omega ^2)^2 + \frac{2D K P_\mathrm{c}\omega ^2}{(1+\gamma P_\mathrm{c})^2}}, \nonumber \\ \end{aligned}$$
(15)

where the subscripts 1, 2 refer to the plus and minus signs, respectively. According to formula (15), a cw field will be unstable if the real part of \(\lambda\) is positive. Therefore, at zero modulation frequency where \(\lambda _1= 0\) and \(\lambda _2=2\frac{\varGamma P_\mathrm{c}}{(1+\gamma P_\mathrm{c})^2} - 2\sqrt{P_\mathrm{c}}\epsilon _\mathrm{c}\), we need \(\epsilon _\mathrm{c} < \frac{\varGamma \sqrt{P_\mathrm{c}}}{(1+\gamma P_\mathrm{c})^2}\) for laser to self-start. Quantitatively this condition suggests two characteristic values of \(P_\mathrm{c}\) above which self-starting can occur: one of them is negative and hence unstable, while the positive one,

$$\begin{aligned} P_\mathrm{c}^{(\gamma )}=\frac{P_\mathrm{c}^{(0)}}{1 - \gamma \sqrt{P_\mathrm{s} g_0/\varGamma }}, \end{aligned}$$
(16)

sets a threshold value of the input field above which the field will be unstable in cw mode. Note that in the case of mode-locked fiber lasers with cubic nonlinearity, this threshold value is obtained as:

$$\begin{aligned} P_\mathrm{c}^{(0)}=P_\mathrm{s}\left( \sqrt{g_0/P_\mathrm{s}\varGamma } - 1\right) . \end{aligned}$$
(17)

To gain a sight on conditions for cw stability in the case of nonzero values of the modulation frequency \(\omega\), the two eigenvalues \(\lambda _{1,2}(\omega )\) were mapped onto a complex parameter space \(\text {Im}[\lambda (\omega )]=F(\text {Re}[\lambda (\omega )])\) where \(\text {Im}[\lambda (\omega )]\) is the imaginary part and \(\text {Re}[\lambda (\omega )]\) is the real part of \(\lambda (\omega )\). Figures 2 and 3 show the complex parameter space for values of the modulation frequency extending in the range \(-5\le \omega \le 5\), values of characteristic parameters of the model are given in the captions.

Fig. 2
figure 2

Parametric plots of \(\lambda _1\) (solid curves) and \(\lambda _2\) (dashed curves) in the complex plane, for \(B=D=0.05\), \(\varGamma =0.1\), \(K=0.01\), \(Ps=2\), \(g_0= 4\), \(T_\mathrm{e}=200\) and different values of \(\gamma\). Here \(P_\mathrm{c}=2\), and SS means “self-starting”

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Fig. 3
figure 3

Parametric plots of \(\lambda _1\) (solid curves) and \(\lambda _2\) (dashed curves) in the complex plane, for \(B=D=0.05\), \(\varGamma =0.1\), \(K=0.01\), \(Ps=2\), \(g_0= 4\), \(T_\mathrm{e}=200\) and different values of \(\gamma\). Here \(P_\mathrm{c}=4\), and SS means “self-starting”

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According to the graphs, self-starting is stronger for the cubic nonlinearity and is enhanced by an increase of \(P_\mathrm{c}\). However, when we increase \(\gamma\) for a fixed value of the input intensity \(P_\mathrm{c}\), the region where \(\text {Re}{(\lambda )}\) is positive reduces, hence reducing the parameter value ranges over which cw are unstable. Actually this later observation is consistent with the dependence of the threshold value \(P_\mathrm{c}^{(\gamma )}\) of \(P_\mathrm{c}\), on the nonlinearity saturation coefficient \(\gamma\) obtained in formula (16) and suggesting a higher input field for laser self-starting when the nonlinearity is of a saturable type.

It is worthwhile recalling that the above analysis assumes the laser will self-start (automatically in the pulse regime) when the cw regime is unstable. On the other hand in our discussions we assumed that the optical system is in a normal dispersion regime (i.e., B and D are positive), where the CGLE is equivalent to the NLSE such that the system admits quasi-Schrödinger sech-type pulses in the mode-locked regime. Still, although negative group-delay dispersion and spectral filter act against NLS sech-type pulses, experiments have demonstrated that pulses can still form in this case. The most interesting experimental evidences have been reported in Ref. [13], where multi-periodic and bound pulse states have been shown to form when B and D cross zero from the positive branch. As the two parameters decrease in the negative branch, multiple-pulse structures sharpen while pulse duration get shorter and shorter. So to say self-starting is also possible in the anomalous dispersion regime, and in the present particular context this is evidenced by the parametric plots of \(\lambda _1\) and \(\lambda _2\) shown in Figs. 4 and 5.

Fig. 4
figure 4

Parametric plots of \(\lambda _1\) (solid curves) and \(\lambda _2\) (dashed curves) in the complex plane, for \(B=D=-0.05\), \(\varGamma =0.1\), \(K=0.01\), \(Ps=2\), \(g_0= 4\), \(T_\mathrm{e}=200\) and different values of \(\gamma\): \(P_\mathrm{c}=2\)

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Fig. 5
figure 5

Parametric plots of \(\lambda _1\) (solid curves) and \(\lambda _2\) (dashed curves) in the complex plane, for \(B=D=-0.05\), \(\varGamma =0.1\), \(K=0.01\), \(Ps=2\), \(g_0= 4\), \(T_\mathrm{e}=200\) and different values of \(\gamma\): \(P_\mathrm{c}=4\)

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4 Nonlinear dynamics

In the previous section, we analyzed the stability of cw solutions to the model Eqs. (1)–(2) following the modulational-instability approach. We obtained that for specific values of model parameters, cws will be unstable causing laser self-starting.

In the modulational-instability picture laser self-starting means that conditions are favorable [11] for the generation of high-intensity pulse patterns. In this section we shall examine profiles of these high-intensity pulse patterns, also paying attention to time variations of the gain g as we change the value of the homogeneous gain \(g_0\). In this purpose we retain a specific nonlinear solution to the CGLE (1), describing an optical field with real amplitude and real phase namely [36]:

$$\begin{aligned} u(z, \tau )=a(\tau )\text {exp}[i\phi (\tau )-i\omega z], \end{aligned}$$
(18)

where a and \(\phi\), which are real functions of \(\tau =t-vz\), represent the amplitude and phase, respectively, of the laser field. The quantity v is the inverse envelope velocity and \(\omega\) is a nonlinear propagation constant. Replacing this solution in Eqs. (1) and (2) and isolating real from imaginary parts we obtain:

$$\begin{aligned} y'= & {} -\frac{Bv}{B^2+D^2}y - \frac{B(g-\ell )+D(\omega +\theta )}{B^2+D^2}a+M^2a \nonumber \\- & {} \frac{Dv}{B^2+D^2}Ma + \frac{(D^2\varGamma -DBK)}{B(B^2+D^2)}\frac{a^3}{1+\gamma a^2}, \end{aligned}$$
(19)
$$\begin{aligned} M'= & {} -\frac{BvM}{B^2+D^2}y - \frac{B(\omega +\theta )}{B^2+D^2}a-\frac{2yM}{a} \nonumber \\+ & {} \frac{Dv}{B^2+D^2}\frac{y}{a} + \frac{D(g-\ell )}{D^2+B^2} + \frac{(D\varGamma -BK)}{B^2+D^2}\frac{a^2}{1+\gamma a^2}, \end{aligned}$$
(20)
$$\begin{aligned} a'= & {} y, \end{aligned}$$
(21)
$$\begin{aligned} \phi '= & {} M, \end{aligned}$$
(22)
$$\begin{aligned} g'= & {} -\frac{(g-g_0)}{T_0}-\frac{a^2}{T_0 P_\mathrm{s}}g, \end{aligned}$$
(23)

where prime symbols denote the derivative with respect to \(\tau\). The five coupled first-order nonlinear ordinary differential equations (19)–(23) were simulated numerically, using an explicit sixth-order Runge–Kutta scheme [39]. In numerical simulations, we fixed characteristic parameters of the model, except the homogeneous gain \(g_0\) which was varied.

Figures 6, 7 and 8 show numerical results of the laser amplitude a, the instantaneous frequency M, the gain g and the amplitude phase plane of laser amplitude for \(g_0=2\), \(g_0=1.2\) and \(g_0=0.8\), respectively. Values of other parameters are given in the captions.

Fig. 6
figure 6

Time series of the laser amplitude a, instantaneous frequency M and gain g, and a–M phase plane, obtained from numerical simulations for \(g_0=2\). Values of parameters are: \(\theta =2\), \(\ell = 0.1\), \(B=0.15\), \(D=0.05\), \(\varGamma =0.4\), \(K=1\), \(\gamma =0.1\), \(T_0=20\), \(P_\mathrm{s}=1\), \(v=0.05\), \(\omega =0.1\)

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Fig. 7
figure 7

Time series of the laser amplitude a, instantaneous frequency M and gain g, and a–M phase plane, obtained from numerical simulations for \(g_0=1.2\). Values of parameters are: \(\theta =2\), \(\ell = 0.1\), \(B=0.15\), \(D=0.05\), \(\varGamma =0.4\), \(K=1\), \(\gamma =0.1\), \(T_0=20\), \(P_\mathrm{s}=1\), \(v=0.05\), \(\omega =0.1\)

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Fig. 8
figure 8

Time series of the amplitude a, instantaneous frequency M and gain g, and a–M phase plane, obtained from numerical simulations for \(g_0=0.8\). Values of parameters are: \(\theta =2\), \(\ell = 0.1\), \(B=0.15\), \(D=0.05\), \(\varGamma =0.4\), \(K=1\), \(\gamma =0.1\), \(T_0=20\), \(P_\mathrm{s}=1\), \(v=0.05\), \(\omega =0.1\)

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On Figs. 6 and 7, the laser amplitude a and instantaneous frequency M are seen to be sparse weakly anharmonic oscillations localized near the origin along the time axis, with the gain g increasing exponentially. As we decrease the value of the homogeneous \(g_0\) the exponential increase of g is gradually softened, below a critical value of \(g_0\) the gain g starts to increase with time and concomitantly, the anharmonic oscillations of the amplitude a and instantaneous frequency M become more and more strong and regular in time, giving rise to a periodic train of pulses (Fig. 8). Instructively, in the simulations we observed that the generation of pulse train was conditioned by relatively small values of the nonlinearity saturation coefficient \(\gamma\). As \(\gamma\) was increased, multi-periodic pulse trains were gradually favored. This last feature was actually expected, given that CGLEs with higher-order nonlinearities (higher than cubic) are known to be prone to multi-periodic pulse trains [26, 36, 37], bound-soliton states [13, 26,27,28,29,30,31,32], optical soliton crystals with multiple periodicities [6, 7], etc.

5 Concluding remarks

We have investigated the dynamics of a model of passively mode-locked fiber laser with saturable absorber, considering that the optical amplifier has a saturable nonlinearity. Starting with cw solutions, we determined conditions under which the laser operation was stable in this regime. More explicitly, by using the modulational-instability analysis we obtained a global stability map which we explored over a relatively broad range of the modulation frequency, for some values of model parameters. Next, seeking for nonlinear solutions, an ansatz was introduced and assumed to describe an optical field with real amplitude and real phase as well as a nonlinear propagation constant. By means of numerical simulations using a sixth-order Runge–Kutta algorithm, we obtained temporal shape profiles of the laser amplitude, instantaneous frequency and gain. It emerged that the laser amplitude was a periodic train of pulses at relatively small values of the homogeneous gain and of the nonlinearity saturation coefficient. For these values, the gain was an increasing function of time. With an increase of the nonlinearity saturation coefficient but keeping small values of the homogeneous gain, the laser amplitude was a multi-periodic pulse train. This last behavior is actually in agreement with a well-known general property of the CGLE with high-order (i.e., higher than cubic) nonlinearities [35,36,37, 40].

The present study is an extension of a previous work done by Chen et al. [11], where the authors studied the self-starting dynamics of a similar model with a cubic nonlinearity. In this previous study the authors did not treat the system dynamics in the nonlinear regime, however it is well known that the CGLE with cubic nonlinearity admits pulse structures [41].