Journal of Computational Methods in Physics
Volume 2014 (2014), Article ID 308947, 13 pages
http://dx.doi.org/10.1155/2014/308947

Research Article

Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation

Ivan M. Uzunov¹ and Zhivko D. Georgiev²

¹Department of Applied Physics, Technical University Sofia, 8 Kl. Ohridski Boulevard, 1000 Sofia, Bulgaria
²Department of Theoretical Electrical Engineering, Technical University Sofia, 8 Kl. Ohridski Boulevard, 1000 Sofia, Bulgaria

Received 31 May 2014; Revised 4 September 2014; Accepted 5 September 2014; Published 15 October 2014

Academic Editor: Ivan D. Rukhlenko

Copyright © 2014 Ivan M. Uzunov and Zhivko D. Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the dynamics of the localized pulsating solutions of generalized complex cubic-quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. Using ansatz of the travelling wave and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability. We apply the Melnikov method also to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification. We prove the existence and stability of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the corresponding unperturbed system. The condition of existence of the limit cycle derived here coincides with the relation between the critical value of velocity and the amplitude of the solitary wave solution (Uzunov, 2011).

1. Introduction

As is well known the complex cubic Ginzburg-Landau equation has been used to describe a variety of phenomena including second-order phase transitions, superconductivity, superfluidity, Bose-Einstein condensation, and strings in field theory [1, 2].

Perturbative methods are used for the description of spatiotemporal pattern formations in systems driven away from equilibrium near the threshold where the nonlinearities are weak and the spatial and temporal modulations of the unstable modes are slow [1]. One of them is the method of “amplitude (model) equations” for the envelope function of unstable mode [1]. Model equations depend on the type of the linear instability. Universal amplitude equations are the real and complex cubic Ginzburg-Landau equation (CCGLE) as well as their generalizations, like the coupled complex cubic Ginzburg-Landau equation and the complex cubic-quintic Ginzburg-Landau equation (CCQGLE). CCGLE describes the evolution of the envelope function of unstable mode for any process exhibiting a Hopf bifurcation. As a model equation CCGLE is applied to the study of oscillatory uniform instability in lasers, oscillatory periodic instability in Rayleigh- Bénard convection in binary mixtures as well as electrohydrodynamic instabilities in nematic liquid crystals (see [1, 2] and references therein).

CGLE also appears as a continuous limit of the amplitude equations that describe the chain of equal weakly nonlinear oscillators with the nearest neighbor interaction. The weakly nonlinear oscillators exhibit a limit cycle.

It has been recently established that in optics the one-dimensional complex cubic-quintic Ginzburg-Landau equation (CCQGLE) can model soliton transmission lines [3, 4] as well as passively mode-locked laser systems [5, 6].

CCQGLE has exact chirped solitary wave solutions [7–10]. Their mathematical nature and the consistent way of derivation have been deeply discussed in [10]. Numerical solutions of CCQGLE could be divided into two groups: localized fixed-shape solutions and localized pulsating solutions. Novel numerical pulsating solutions of the CCQGLE, namely, pulsating, creeping, snaking, and erupting solutions have been reported in [11]. Chaotic pulsating solutions and period doubling were reported in [12]. Detailed review of the analytical and numerical solutions of CCQGLE can be found in [13–15].

A theoretical approach for analysis of the observed solutions of CCQGLE using the variational method has been reported [16, 17]. The resulting Euler-Lagrange equations have been analyzed for the existence of periodic, quasiperiodic, and chaotic attractors [16, 17]. It has been shown that the different numerically observed solutions of CCQGLE (dissipative solitons) may be related to the stable periodic attractors of the Euler-Lagrange equations [16, 17].

The influence of the higher order effects, namely, the third-order of dispersion (TOD), intrapulse Raman scattering (IRS), and self-steeping effect (see for detailed description these effects [18–20]) on the description of fiber laser operation has been studied in [21]. In order to perform this analysis we consider a generalized CCQGLE that includes the higher-order effects [21]. The existence of the exact chirped solitary solution of this generalized CCQGLE has been reported for the first time in [22]. Very recently, the influence of these high order effects on the dynamics of pulsating, erupting, and creeping solutions using the generalized CCQGLE has been studied in [23, 24]. Generally, it has been shown that these higher-order effects can have strong impact on these solutions. It was established that under the influence of IRS and TOD, the plain pulsating and the creeping solutions can lose their pulsating behavior [23]. A further observation was made that in the presence of all higher-order effects the explosions of an erupting soliton can be reduced and even eliminated [23, 24]. Numerical findings of [23] suggest that in the presence of IRS the pulsating regime is stabilized (see Figures 2(a) and 2(d) in [16]), or, in other words, a periodic attractor in (1) appears.

The main aim of this paper is to examine analytically the influence of the intrapulse Raman scattering on the localized pulsating solutions of the generalized CCQGLE in an attempt to explain the very recent numerical results of [23]. In fact we propose a theoretical approach for identification of periodic attractors of (1). We introduce a dynamical system with finite degrees of freedom or a system of ordinary differential equations (SODE) related to generalized CCQGLE. We next identify the periodic attractors of this SODE. The last step of this approach is to solve numerically the generalized CCQGLE in order to compare periodic attractors of SODE with those of CCQGLE (this step will be the subject of further investigation).

In order to introduce SODE, we use ansatz of the travelling wave. Fixing some relations between the material parameters of (1), we have succeeded to derive the equation of strongly nonlinear Lienard-Van der Pol oscillator (see (11)) for the amplitude of the nonlinear wave. After identifying the possible equilibrium points of (11) in the general case, we had to fix some values of his coefficients in order to demonstrate our approach. The simpler version of the equation of strongly nonlinear Lienard-Van der Pol oscillator, namely, the equation of strongly nonlinear Duffing-Van der Pol oscillator, has been the object of intensive study by means of different perturbation methods [25]. Interestingly enough, even for the large values (larger than 1) of the small parameter, appearance of stable periodic attractors-limit cycles was observed [25]. Here we apply the Melnikov method [26–29] to analyze the possible existence of limit cycles in the equation of strongly nonlinear Lienard-Van der Pol oscillator and the equation of strongly nonlinear Duffing-Van der Pol oscillator in two cases, around closed oval trajectories having finite temporal period and around homoclinic trajectory with infinite temporal period. For the fixed values of the coefficients of the equation of strongly nonlinear Lienard-Van der Pol oscillator we prove the existence of a single limit cycle that arises around a closed phase trajectory of the unperturbed system. We could then expect that the observed limit cycle will be related to the corresponding periodical attractor of (1).

The second aim of this paper is to explore analytically the influence of the intrapulse Raman scattering on the localized pulsating solutions of the particular case of generalized CCQGLE, namely, (1) with . Physically this equation describes the bandwidth limited amplification and has recently been investigated in [30, 31]. Equation of the strongly nonlinear Duffing-Van der Pol oscillator has been introduced [30] in order to describe the influence of IRS on the soliton solutions. Here we apply the Melnikov method to the equation of strongly nonlinear Duffing-Van der Pol oscillator and prove the existence of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the unperturbed system. Due to the large period of periodic movement, the limit cycle transforms into perturbed soliton solution. The condition of existence of the limit cycle derived here coincides with the condition of existence of perturbed soliton solution derived by means of the soliton perturbation theory in [30] and the critical value of the homoclinic bifurcation parameter calculated in [30] by means of the hyperbolic perturbation method [32] and hyperbolic Lindstedt-Poincare method [33]. As a result of [30] which has been successfully confirmed by the numerical solution of the corresponding generalized CCQGLE in [31], we can expect that the obtained results here for (1) will also well correlate to the numerical solution of (1).

The results presented here have been first announced (but not published) in [34] and in a brief and different form reported in [35].

The paper is organized as follows. First, the physical meaning and applications of the generalized CCQGLE are presented in Section 2. Second, the derivation of the equation of strongly nonlinear Lienard-Van der Pol oscillator (see (10) below) is given in Section 3. Next, in Section 4 the results of the analysis of (10) by means of the Melnikov method are given. In Section 5 we apply the Melnikov method to the investigation of the influence of the IRS on the bandwidth limited amplification [30]. In Section 6 we discuss the obtained results. Finally, we make our conclusions in Section 7. A classification of the equilibrium points of (10) is given in Appendix A. Appendix B contains a description of the calculation of Abelian integrals necessary for the application of the Melnikov method. Appendix C gives the condition of existence of perturbed solitary wave solution [30] derived by the hyperbolic perturbation method and the hyperbolic Lindstedt-Poincare method as well as the relationship between the equilibrium amplitude and the velocity of the perturbed soliton solution, obtained by means of soliton perturbation theory [30].

2. Basic Equation

The propagation of ultrashort pulses in the presence of spectral filtering, linear and nonlinear gain/loss, and intrapulse Raman scattering is described by the following generalized CCQGLE [23, 24]:

where is the normalized envelope of the electric field, is the normalized propagation distance, is the retarded time, is the linear gain or loss coefficient, describes spectral filtering (gain dispersion), and is related to nonlinear gain-absorption process, , if negative accounts for the saturation of the nonlinear gain, , if negative corresponds to the saturation of the nonlinear refractive index. The last term in (1) describes the intrapulse Raman scattering (IRS)—the important nonlinear physical effect which has to be taken into account when femtosecond optical pulses propagate in optical fibers. It is related to the first moment of the nonlinear response function (the slope of the Raman gain spectrum) and leads to new physical phenomena as soliton self-frequency shift and breakup of the N-soliton bound states (for the review see [18–20]).

Equation (1) has been used to model solving of the problem of linear-wave growth in bandwidth-limited-amplified soliton transmission systems [3, 4]. In the context of solid state lasers, (1) () has been proposed for description of a fast saturable absorber and additive pulse mode-locking [5] and later has been used () as a model for self-limited additive pulse mode-locking [36]. Next it turned out that (1) () was also applicable as a main equation for the mode-locked fiber lasers [37]. Recently the all-normal-dispersion passively mode-locked fiber lasers have been successfully described by means of (1) () [38, 39]. Generally, the CCQGLE () has proved to be a good model for the real mode-locked lasers (for the review see [40, 41]).

As a result of intensive numerical investigation of CCQGLE, some areas in the space of physical parameters have been established, in which there exist stable localized solutions of CCQGLE [14, 42].

3. Derivation of Lienard-Van der Pol Equation

Now we will look for the stationary pulse solutions of (1) in the form:

where and and are real numbers. has a meaning of the unknown inverse equilibrium velocity. Inserting (2) into (1) the following nonlinear system of ordinary differential equations for the amplitude and phase functions and is obtained:

Summing up (3a) multiplied with and (3b) we obtain

Multiplying (4) with we get

Assume the following conditions are satisfied:

and introduce the function

Equation (5a) can be written in the form, , with corresponding solution: .

The phase function then is completely determined by the amplitude function :

If we replace with (7) in (3a), the following equation for the amplitude function is obtained:

Let us introduce the following quantities:

The coefficients , , , , , depend on , , , . Using the quantities (9) and replacing and as well as by (5b), (8) can be written in the form:

As we can see from (10), by increasing , the role of the terms proportional to and will decrease. The constant can be freely chosen to be smaller than unity. Then we could expect that the terms proportional to and can be safely neglected, in case we consider the dynamics of (10) at sufficiently long distances . Neglecting the terms proportional to and we obtain

Equation (11) is the equation of a strongly nonlinear Lienard-Van der Pol oscillator.

In order to check the correctness of the approximation of (10) with (11), we have numerically solved the systems corresponding to these equations for some fixed values of parameters and compared the obtained results. As the aim of this work is to find the limit cycles, we should compare the abilities of (11) to describe the limit cycles predicted by (10).

We have found that for the values of quantities , , , , , of order of unity, and , for properly long times both systems give comparable results. The difference appears at the initial stage and it is proportional to the value of . For very small values of (), the difference is so small that it can be safely neglected. For values of and smaller than unity, the acceptable value of increases.

In order to demonstrate these results we fix the following values of the coefficients , , , , , and . The values of , , , , are determined by the values of , , , , , given in Section 6. The value of is chosen to satisfy the inequality given by (26). In this case according to our analysis performed in Section 4, (11) possesses a stable limit cycle. In order to demonstrate the existence and stability predicted by (11) limit cycle, we compare the results of the numerical integration of systems that correspond to (10) and to (11) for two different initial conditions: (a) , , and (b) , . The obtained results for functions and are shown in Figures 1(a) and 1(b) and Figures 2(a) and 2(b), respectively.

(a)

(b)

Figure 1: Evolution of (a) and (b), according to (9) (blue) and (10) (red), for , , , , , , . Initial conditions: , .

(a)

(b)

Figure 2: Evolution of (a) and (b), according to (9) (blue) and (10) (red), for , , , , , , . Initial conditions: , .

As can be seen from Figures 1(a) and 1(b), the discrepancy between the two equations appears at the initial stage (till ). For larger the behavior of both functions is very similar. The periodic behavior of the functions and shows the existence of the stable limit cycle.

As can be seen from Figures 2(a) and 2(b), the behavior of both functions with is quite similar. Again, the periodic behavior of the functions and shows the existence of the stable limit cycle. Comparing the results shown in Figures 1(a) and 1(b) and Figures 2(a) and 2(b), we see that the stable limit cycle predicted by (11) also exists for (10). So, we believe that the approximation of (10) with (11) is acceptable in a proper region of parameters.

From the definition of it follows that and . In our further investigation we will assume that these inequalities are satisfied. Each of the coefficients and can be either positive or negative. Moreover, we will assume that the right hand side of (11), which can be regarded as a dissipative term, is a small quantity, or small perturbation. Then the unperturbed equation corresponding to (11) is exactly the so-called cubic-quintic Duffing equation. Many methods for the analysis of (11) require in advance that we obtain the phase portrait of the cubic-quintic Duffing equation. For this reason a classification of the equilibrium points of this equation is given in Appendix A.

4. The Melnikov Method for Analysis of Lienard-Van der Pol Equation

Consider (11) under the following conditions: , and . After applying the scaling

(11) takes the form

where

Further we assume that is a small parameter. Then the right hand side of (13) is a small perturbation. Taking into account the expression for given in (9), we can conclude that is a small parameter on condition that the spectral filtering has a small value, or the module of linear gain or loss coefficient has a large value, or when both conditions are fulfilled.

Our goal is to establish that (13) admits limit cycles and to analyze them. For this purpose we will use two main mechanisms for the arising of limit cycles: (i) arising of limit cycles around a closed trajectory belonging to a family of such closed trajectories of the unperturbed equation; (ii) arising of limit cycles around a homoclinic trajectory of the unperturbed equation. The limit cycles in the first case correspond to a periodic solution in the initial wave equation. The limit cycle around a homoclinic trajectory in the second case (with infinitely long period) corresponds to a soliton solution in the initial wave equation.

The analysis of limit cycles will be carried out using the Melnikov theory [27], which also allows finding the conditions for the arising of limit cycles. A classification of the equilibrium points of the unperturbed cubic-quintic Duffing equation depending on the values of its coefficients is given in Appendix A for convenience. In this situation, one can easily obtain the phase portrait of the unperturbed equation and determine whether the preliminary requirements of the Melnikov theory (presence of a family of closed trajectories, or presence of homoclinic trajectories) are fulfilled. In order to be able to perform the computations and to get a better representation of the results we choose . If we choose another value for , the phase portrait of the unperturbed cubic-quintic Dufing equation is changed, but when the preliminary requirements of the Melnikov theory are fulfilled, the method of study of the limit cycles remains the same. Only the calculations leading to other limit cycles with other parameters are changed.

Taking into account the value of , (13) can be written as the following system:

where the dot means differentiation with respect to . The perturbation functions in this system are

The unperturbed system (i.e., system (15) with is

The last system is Hamiltonian with Hamiltonian function

where the parameter corresponds to a constant Hamiltonian level.

System (17) has the following equilibrium points, , which is a “saddle point,” , which are “centers,” and , which are “saddle points.” The phase portrait of system (17) is symmetrical concerning - and -axis and is shown in Figure 3. Note that this phase portrait corresponds to Subcase 6.1 given in Appendix A.

Figure 3: Phase portrait of the unperturbed Hamiltonian system (17). The five equilibrium points, two centers (green points) and three saddle points (red points), can be clearly observed. The homoclinic trajectory forms figure “eight loop.”

According to the Melnikov theory, limit cycles in a given perturbed Hamiltonian system can arise around a closed phase trajectory of the unperturbed Hamiltonian system. Because of this, we will further be interested only in the family of closed trajectories localized within the right half of the figure “eight loop,” that is, the closed trajectories localized between the center and the homoclinic loop. Each phase trajectory corresponds to a constant value of the Hamiltonian function; that is, , . The value of the Hamiltonian function on the homoclinic trajectory is and in the “center” the value is . It can be easily calculated that the homoclinic trajectory intersects the positive part of the -axis at the point . A given closed trajectory localized within the right half of the figure “eight loop” has the following equation:

Having these notions in mind, we can obtain the Melnikov function for the system (15):

Introducing the functions

the Melnikov function and its derivative with respect to can be written as

Note that and are Abelian integrals.

We will look for the zeros of with . To find these zeros we need additional investigations of the functions , , and . The facts necessary for this purpose are collected in Lemma B.1 presented in Appendix B. A crucial factor in this consideration is the fact that the function is positive and strictly monotone decreasing in the interval . The graph of the function obtained by numerical integration is shown in Figure 4.

Figure 4: The graph of the function obtained by numerical integration of the Abelian integrals given by with the help of Mathematica.

A given zero of the Melnikov function satisfies the equations

Now we can make some conclusions. Since is a strongly monotone decreasing function, the second equation in (25) can have only one zero. The function satisfies the inequality . Therefore the Melnikov function has a single zero in the case when the following inequality is satisfied:

In this case the system (15) has a limit cycle , which is localized in -neighborhood of the closed curve .

According to the Melnikov theory, the stability of the limit cycle is determined by the sign of quantity : stable for and unstable for [27]. From (24) and Lemma B.1 in Appendix B, there follows that

Taking into account the inequality , we conclude that the limit cycle is stable.

Figure 5 shows the limit cycle of system (15) obtained for and .

Figure 5: The phase portrait of system (15) obtained by numerical solution.

Figure 5 presents the results from the calculation of three different initial conditions for the time . One of the initial conditions coincides with the limit cycle and the corresponding phase trajectory stays the same during the calculation. Another initial condition is inside the limit cycle and with the increase of time the trajectory approaches the limit cycle from the inside. The last initial condition is outside the limit cycle and with the increase of time the trajectory approaches the limit cycle from the outside.

The presented theory allows us to synthesize a system of the type (15), having in advance assigned limit cycle. Let be the Hamiltonian level around which we want to a limit cycle to arise. After computing , we get from (25)

The obtained in this way value of provides system (15) with the desired limit cycle.

5. The Melnikov Method for Analysis of Duffing-Van der Pol Equation

In the previous section it was shown that, under certain conditions, (11) has a simple stable limit cycle. This limit cycle arises around a closed phase trajectory of the unperturbed system, which is an oval curve. In the present section we will show that (11) can have a limit cycle that arises in a neighborhood of a homoclinic trajectory of the unperturbed system. In the first case, the closed oval curve, respectively, the emerging limit cycle, has a finite period. In the second case, as is well known, the homoclinic trajectory is described by the solution of the unperturbed equation for infinitely long time. On account of this, the limit cycles in the two cases have different properties and this leads to different behaviors of the initial wave equation.

The process of arising of limit cycles in a perturbed Hamiltonian system around a homoclinic trajectory of the unperturbed Hamiltonian system is considered for the first time by Andronov; see Andronov et al. [26] (therefore this process is called Andronov mechanism). Later this theory was developed by Roussarie and shown in a form convenient for our applications in [43]. Useful and detailed information on these subjects is given in the book of Han and Yu [44].

Formally the analysis below will be performed for the case , , and . (Note however, that , in this paragraph are short notations of the magnitudes , defined in Appendix C.) This case has been studied earlier in [30] by means of the hyperbolic perturbation method [32] and the hyperbolic Lindstedt-Poincare perturbation method for homoclinic motion [33].

After applying the scaling

(10) takes the form

where

Further, we will assume that is a small parameter. (Having in mind the definition of in Appendix C this requires that the parameter be sufficiently large.)

Equation (30) can be written as a perturbed Hamiltonian system in the following way:

where the dot means differentiation with respect to . The perturbation functions in this system are

The unperturbed system (with ) is Hamiltonian with Hamiltonian function

Moreover, the unperturbed system has three equilibrium points—a hyperbolic “saddle point” and two nondegenerate “centers” . The value of the Hamiltonian function at the two “centers” is . It is necessary to note that the point is also an equilibrium point “saddle point” for the perturbed system (32).

The equation defines a symmetric double homoclinic loop (figure eight-loop), consisting of two homoclinic orbits and connecting the “saddle point” to itself; that is, . The homoclinic orbits and are localized, respectively, in the half planes and . The solution of the unperturbed system along the homoclinic orbits and can be expressed in the time domain in the following way:

On the other hand, the equation defines three continuous families of periodic orbits—, , and . The periodic orbits and surround the centers and tend, respectively, to and as . The limit of as is the double homoclinic loop .

In general, the Melnikov function can be defined on each of the mentioned three families of periodic orbits and we have three Melnikov functions [44]. Let us introduce the Melnikov function for the perturbed Hamiltonian system (32) defined along the periodic orbit . It is well known [43, 45] that the Melnikov function has the following asymptotic expansion near (, or )

where

The quantity is the Melnikov function computed along the separatrix (the computation along is identical and leads to the same results) and for the system (32) it has the form

The coefficients , , ,… play an important role in the study of the limit cycle bifurcation. According to a theorem of Roussarie [43], if the following relations hold

then for small the system (32) can have the greatest number of limit cycles in a small neighborhood of the homoclinic loop.

In our case it is easy to compute . Thus the equation gives the condition for which in system (32) a single limit cycle bifurcates from the homoclinic loop of the unperturbed system. Taking into account (32), (33), (35a), (35b), and (37) and having conducted some straightforward calculations, we obtain

Hence

Equation (40) expresses the condition for arising of two homoclinic orbits and connecting the “saddle point” to itself in system (32). It can be said, that under conditions (40), the perturbed Hamiltonian system (32) admits a single homoclinic limit cycle , consisting of the orbits and ; that is, . Let us remember that the perturbations in system (32) preserve the equilibrium point unchanged. The limit cycle is an isolated homoclinic loop located in a small neighborhood of the loop and passing through the point .

Further, the stability of the limit cycle is determined by the sign of the saddle point quantity [46]

This means that the separatrix cycles and , as well as the limit cycle , are stable.

Having in mind the derivation of in Appendix C it is clear that the obtained here (40) precisely coincides with condition derived in [30]. From this condition follows the relation between the critical value of velocity and the amplitude of the solitary wave solution related to the homoclinic loop given by . (Note that in (35a), (35b) ). However, as has been established in [30], by means of the soliton perturbation theory the relation between the equilibrium amplitude and velocity of the perturbed soliton solution (see ) in fact coincides with the same relation in . Consequently, the numerical confirmation of (40) and therefore of coincides with the numerical confirmation of .

6. Discussion

Since (11) is derived from (1), we expect that the existence of this single limit cycle of (11) which arises around a closed phase trajectory of the unperturbed system will lead to the existence of periodical attractor of (1). As in this case the emerging limit cycle has a finite period, we expect that the corresponding periodic attractors of the generalized CCQGLE will represent the localized pulsating solutions of this equation. We should mention however that the dynamical system given by (11) possesses different equilibrium points depending on the values of the parameters , we may expect the existence of a variety of limit cycles. Even in the case considered here the other limit cycles could be found in the region between homoclinic and heteroclinic trajectories, or in the neighborhood of a homoclinic or heteroclinic trajectory of the unperturbed system. It is clear that the question of how many limit cycles exist in (11) and the question of their type require further systematic mathematical investigation.

Let us now discuss the question of numerical confirmation of the results obtained in Sections 4 and 5. At this point we do not have any numerical confirmation of the results presented in Section 4. We have fixed the value of , assuming, that , , . As a result, we have obtained the following values of the parameters of (1): , , , , . The application of the Melnikov method leads to additional condition for given by (26). What we can say at this point is that the parameters , , , are in regions for which stable localized fixed-shape solutions and localized pulsating solutions of (1) are identified in [14], while the values of and are larger than the usual values [14, 16]. In order to confirm the limit cycles predicted by our approach in Section 4, we plan a further numerical investigation of the CCQGLE.

In Section 5 the situation is completely different. Having applied the Melnikov method to the equation of Duffing-Van der Pol oscillator we have proved the existence of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the unperturbed system (see (40)). This result coincides with the condition derived earlier in [30]. As the homoclinic trajectory is described by the solution of the unperturbed equation for infinitely long time, the condition of existence of the limit cycle in (40) gives the relation between the critical value of velocity and the amplitude of the solitary wave solution (see as well as (17) in [30]). As has been mentioned in Section 5, there appears a problem in the numerical confirmation of and .

To prove the predicted relation between the critical value of velocity and the amplitude of the solitary wave solution or of the equilibrium values of the amplitude and velocity of perturbed soliton solution , a detailed numerical investigation has been performed in [31]. Equation (1) with has been numerically solved by means of the split-step Fourier method, applying the Blow-Wood RK4 scheme for the following values of parameters , , and [31]. The obtained results in [31] confirm with an excellent accuracy the relation between the equilibrium values of the amplitude and velocity of perturbed soliton solution and therefore of and finally (40).

Taking into account the excellent numerical confirmation of results of Section 5 in [31], we could expect the numerical observation of the limit cycles for the strongly nonlinear Lienard-Van der Pol equation obtained for the first time here in Section 4.

7. Conclusion

We have studied the dynamics of the localized pulsating solutions of generalized cubic-quintic complex Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering.

The main result of this work is a proposal for an approach for identification of periodic attractors of the generalized CCQGLE. First, we use ansatz of the travelling wave and determine some conditions for the material parameters; then we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation and we show that for a set of fixed material parameters a limit cycle arises around a closed phase trajectory of the unperturbed system. After that we prove its stability. Due to the complexity of the strongly nonlinear Lienard-Van der Pol equation, however, it is clear that the question of how many limit cycles exist as well as the question of their type will require further systematic mathematical investigation.

We next have shown that the Melnikov method could be applied to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification [30]. We have proved the existence and stability of a limit cycle that arises in a neighborhood of the homoclinic trajectory of the corresponding unperturbed system. As this trajectory is described by the solution of the unperturbed equation for infinitely long time, the condition of existence of the limit cycle derived in (40) is equivalent to the relation between the critical value of velocity and the amplitude of the solitary wave solution found in [30] (see (17)).

Appendices

A. Equilibrium Points of the Cubic-Quintic Duffing Equation

Consider the following cubic-quintic Duffing equation:

This equation is equivalent to the system

Under the condition , we will receive the equilibrium points for the system and their type in the case where the system is nondegenerate. In general, equilibrium points are given by the following expressions:

It is only necessary for the coordinates obtained from these expressions to be real quantities. The type of equilibrium points is determined by the eigenvalues of the matrix of linearized system corresponding to . Having this in mind, we can make the following classification of the equilibrium points of system depending on the coefficients.

Case 1 (, , ). System has a single equilibrium point—, which is a “saddle point.”

Case 2 (, , ). System has a single equilibrium point—, which is a “center.” Note that from the inequality there follows that the coefficients and have the same signs.

Case 3 (, , ). System has a single equilibrium point—, which is a “center.”

Case 4 (, , ). System has 3 equilibrium points which are

Case 5 (, , ). System has 3 equilibrium points which are

Case 6 (, , ). System has 5 equilibrium points which are

It is interesting to note that, in Case 6, while keeping the number and type of equilibrium points, there are three different phase portraits. These phase portraits are obtained under the following conditions.

 Subcase 6.1: , , . Subcase 6.2: , , . Subcase 6.3: , , .

B. Analysis of the Auxiliary Functions , , and

The basic properties of the functions , , and are collected in the following Lemma.

Lemma B.1. The following statements are valid:

(a)

(b)

(c)

(d)

(e)

(f)

(g)the function is positive and strictly monotone decreasing in the interval , whereupon its derivative is negative; that is, for , and therefore .

We will briefly mention the proofs of these statements. More information can be found in [44, 46–48]. Statements (a) and (e) are proved by using Green’s theorem and mean value theorem. Statement (b) follows from the relations , where is the period (with respect to time) of the closed trajectory . Statements (c), (d), and (f) are proved by direct calculation of integrals (with ), . Finally, statement (g) for monotonicity of the function is proved by using the method of Li and Zhang given in [49].

We have calculated the function numerically. From the Hamiltonian function there follows that

Then the Abelian integrals are given by

The boundaries of integration and represent the crossing point of the curve with the axis . For the values of in the interval we have calculated the following:

(1)the angle ,

(2)the boundaries of integration in the integrals :

(3)integrals given by ;(4)the function .

C. Earlier Application of Duffing-Van der Pol Equation in Description of the Perturbed Soliton Solution

In this appendix we review the earlier results on the application of the Duffing-Van der Pol equation for description of perturbed solitary wave solution in the presence of IRS and bandwidth limited amplification [30].

The Duffing-Van der Pol equation has been considered in [30] in the form (see (11) of [30]):

where , , , , and is a small parameter [30]. and have the same meaning as in (2). The amplitude of the perturbed solitary wave solution is given by .

It has been shown (see (16) in [30]) that if

the velocity (frequency) and the amplitude of the perturbed solitary wave solution are related through the relation (see (17) in [30]):

These results have been obtained by means of the hyperbolic and hyperbolic Lindstedt-Poincare perturbation methods for homoclinic motion proposed, respectively, in [32, 33].

By applying the soliton perturbation theory, practically the same relationship has been derived as the one given by between the equilibrium soliton amplitude and velocity (see (5) in [30]):

Assuming that and , we get . Equation can be then written as

Formally is precisely the same as (11) with , which is discussed in Section 5, but all coefficients are with a hat.

By means of quantity , the condition can be written as

coincides with (40) from Section 5. In conclusion, applying the Melnikov method in the study of the Duffing-Van der Pol equation in Section 5, we have obtained the precise condition derived earlier in [30]. From this condition there follows the relation between the critical value of velocity and the amplitude of the solitary wave solution related to the homoclinic loop given by [30].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are thankful to the referees for their questions and suggestions. They believe that the work done in order to answer their requirements has led to a significant improvement of the paper. One of them, Ivan M. Uzunov, would like to thank Professor Fedor Mitschke for the useful discussion on the initial variant of this paper during the International Conference of Computational Methods in Science and Engineering—ICCMSE, 04–07 April 2014, Athens, Greece.

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