Research Article Localized Pulsating Solutions of the Generalized Complex

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1 Computational Methods in Physics, Article ID , 13 pages Research Article Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation Ivan M. Uzunov 1 and Zhivko D. Georgiev 2 1 Department of Applied Physics, Technical University Sofia, 8 Kl. Ohridski Boulevard, 1000 Sofia, Bulgaria 2 Department of Theoretical Electrical Engineering, Technical University Sofia, 8 Kl. Ohridski Boulevard, 1000 Sofia, Bulgaria Correspondence should be addressed to Ivan M. Uzunov; ivan uzunov@tu-sofia.bg Received 31 May 2014; Revised 4 September 2014; Accepted 5 September 2014; Published 15 October 2014 Academic Editor: Ivan D. Rukhlenko Copyright 2014 I. M. Uzunov and Z. 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. 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 AsiswellknownthecomplexcubicGinzburg-Landauequation 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 areweakandthespatialandtemporalmodulationsofthe 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 theevolutionoftheenvelopefunctionofunstablemode 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 onedimensional 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,

2 2 Computational Methods in Physics namely, pulsating, creeping, snaking, and erupting solutions have been reported in [11]. Chaotic pulsating solutions and period doubling were reported in [12]. Detailedreview 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 higherorder 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] suggestthat inthepresenceofirsthepulsatingregimeisstabilized(see Figures 2(a) and 2(d) in [16]),or,inotherwords,aperiodic 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) withε=μ=] =0.Physicallythis 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 thesolitonsolutions.hereweapplythemelnikovmethodto 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, thederivationoftheequation of strongly nonlinear Lienard-Van der Pol oscillator (see (10) below)isgiveninsection 3. Next,inSection 4 the results of the analysis of (10) by means of the Melnikov method are given. In Section 5we apply the Melnikov method to the investigation of the influence of the IRS on the bandwidthlimitedamplification [30]. InSection 6 we discuss theobtainedresults.finally,wemakeourconclusionsin Section 7. A classification of the equilibrium points of (10) is given in Appendix A. Appendix B contains a description

3 Computational Methods in Physics 3 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]: i U x U 2 t 2 + U 2 U =iδu+iβ 2 U t 2 +iε U 2 U ] U 4 U +iμ U 4 U+γU t ( U 2 ), where U isthenormalizedenvelopeoftheelectricfield, x is the normalized propagation distance, t is the retarded time, δ is the linear gain or loss coefficient, β describes spectral filtering (gain dispersion), and ε is related to nonlinear gain-absorption process, μ, 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) (] = μ = γ = 0) has been proposed for description of a fast saturable absorber and additive pulse mode-locking [5] and later has been used(γ = 0)as a model for self-limited additive pulse mode-locking [36]. Next it turned out that (1)(] =μ=γ=0) 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)(] =γ=0) [38, 39]. Generally, the CCQGLE (γ =0)hasprovedtobea goodmodelfortherealmode-lockedlasers(forthereview see [40, 41]). As a result of intensive numerical investigation of CCQGLE, some areas in the space of physical parameters (δ,β,ε,μ,]) have been established, in which there exist stable localized solutions of CCQGLE [14, 42]. (1) 3. Derivation of Lienard-Van der Pol Equation Now we will look for the stationary pulse solutions of (1) in the form: U (x, t) =u(ξ) exp (i (f (ξ) +Kx)), (2) where ξ = t Mx and M and K are real numbers. M 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 functionsu(ξ) and f(ξ) is obtained: βu +f u Mu δu+βu(f ) 2 +( 1 2 )uf εu 3 μu 5 =0, ( 1 2 )u +2βu f 2γu 2 u Ku+Muf ( 1 2 )u(f ) 2 +βuf +u 3 + ]u 5 =0. (3a) (3b) Summingup (3a)multipliedwith1/(2β) and (3b)we obtain (2δ+4βK)u+4βMuf +(1+4β 2 )uf +2(1+4β 2 )u f (2M + 8γβu 2 )u +(4β 2ε)u 3 +(4]β 2μ) u 5 =0. Multiplying (4)withu(ξ) we get d dξ [(1 + 4β2 )u 2 f Mu 2 2βγu 4 ] +[ (2δ+4βK)u 2 +4βMu 2 f +(4β 2ε)u 4 +(4]β 2μ) u 6 ]=0. Assume the following conditions are satisfied: 1+4β 2 =4Mβ; ε=β(γ+2); andintroducethefunction M=2(δ+2βK); μ=2β], (4) (5a) (5b) φ (ξ) =u(ξ) 2 [ M + (1 + 4β 2 )f (ξ) 2βγu(ξ) 2 ]. (6) Equation (5a) canbewrittenintheform,φ (ξ) + φ(ξ) = 0, with corresponding solution: φ(ξ) = φ 0 e ξ, φ 0 = const. The phase function f(ξ) then is completely determined by the amplitude function u(ξ): f (ξ) = M (1 + 4β 2 ) ξ+ φ 0 (1 + 4β 2 ) e ξ u 2 dξ + 2βγ (1 + 4β 2 ) u2 dξ. (7)

4 4 Computational Methods in Physics If we replace f(ξ) with (7)in(3a), the following equation for theamplitudefunctionu(ξ) is obtained: ξ γβ u +( δ β M 2 (1 + 4β 2 ) 2 4φ 0e (1 + 4β 2 ) 2 ) u = +( ε β 4Mγβ (1 + 4β 2 ) 2 )u3 +( μ β 4γ 2 β 2 (1 + 4β 2 ) 2 )u5 4 (1 + 4β 2 ) ( βm + γu2 )u. Let us introduce the following quantities: c 1 = c 3 =2+ φ 2 0 e 2ξ (1 + 4β 2 ) 2 1 u δβ 4βγ 16β 2, c 2 = (1 + 4β 2 ), 4γβ 2 (1 + 4β 2 ), c 1 4 = (1 + 4β 2 ) 2, c 5 = μ β 4γ 2 β 2 (1 + 4β 2 ) 2, σ = 4γ (1 + 4β 2 ). The coefficients c 1, c 2, c 3, c 4, c 5, σ depend on δ, β, μ, γ. Using the quantities (9) and replacing K and M as well as ε by (5b), (8)canbewrittenintheform: u +(c 1 c 2 φ 0 e ξ ) u+c 3 u 3 +c 5 u 5 c 4 φ 2 0 e 2ξ u 3 =( 1+σu 2 )u. (8) (9) (10) As we can see from (10), by increasing ξ, theroleofthe terms proportional to φ 0 e ξ and φ 2 0 e 2ξ will decrease. The constant φ 0 canbefreelychosentobesmallerthanunity. Then we could expect that the terms proportional to c 2 and c 4 canbesafelyneglected,incaseweconsiderthedynamics of (10) at sufficiently long distances ξ. Neglecting the terms proportional to c 2 and c 4 we obtain u +c 1 u+c 3 u 3 +c 5 u 5 =( 1+σu 2 )u. (11) 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 parametersandcomparedtheobtainedresults.astheaimof 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 c 1, c 2, c 3, c 4, c 5, σ of order of unity, and φ 0 0.1, forproperlylongtimes both systems give comparable results. The difference appears at the initial stage and it is proportional to the value of φ 0.For very small values of φ 0 (φ 0 1), the difference is so small that itcanbesafelyneglected.forvaluesofc 2 and c 4 smaller than unity, the acceptable value of φ 0 increases. In order to demonstrate these results we fix the following values of the coefficients c 1 = 1, c 2 = 1.005, c 3 = 5/2, c 4 = , c 5 = 1,andφ 0 = 0.1.Thevaluesofc 1, c 2, c 3, c 4, c 5 are determined by the values of δ = , β = , ε = , μ = , ] = , γ = given in Section 6.Thevalueofσ= 2.15 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) u(0) = 0.2, du/dξ(0) = 0, and (b) u(0) = 1.0, du/dξ(0) = 0.Theobtained results for functions u(ξ)and du/dξ(ξ) are shown in Figures 1(a) and 1(b) and Figures 2(a) and 2(b),respectively. As can be seen from Figures 1(a) and 1(b),thediscrepancy between the two equations appears at the initial stage (till ξ 15). For larger ξ the behavior of both functions is very similar. The periodic behavior of the functions u(ξ) and du/dξ(ξ) 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 ξ is quite similar. Again, the periodic behavior of the functions u(ξ) and du/dξ 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 γ > 0 it follows that c 3 > 0 and σ > 0. In our further investigation we will assume that these inequalities are satisfied. Each of the coefficients c 1 and c 5 canbeeitherpositiveornegative.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: c 1 <0, c 3 >0 and c 5 <0. After applying the scaling (11)takestheform d 2 X dτ 2 u= 4 c 1 c 5 X, ξ = 1 c 1 τ, (12) X+ c 3X 3 X 5 = ε( 1+ bx 2 ) dx dτ, (13)

5 Computational Methods in Physics u du/dξ ξ (a) ξ (b) Figure 1: Evolution of u(ξ) (a) and du/dξ (b), according to (9) (blue) and (10)(red),forc 1 = 1, c 2 = 1.005, c 3 =5/2, c 4 = , c 5 = 1, σ= 2.15, φ 0 = 0.1. Initial conditions: u(0) = 0.2, du/dξ(0) = u 0.6 du/dξ ξ ξ 0.6 (a) (b) Figure 2: Evolution of u(ξ) (a) and du/dξ (b), according to (9) (blue) and (10)(red),forc 1 = 1, c 2 = 1.005, c 3 =5/2, c 4 = , c 5 = 1, σ= 2.15, φ 0 = 0.1. Initial conditions: u(0) = 1.0, du/dξ(0) = 0. where c 3 = ( c 3 c 1 ) c 1 c 5 >0, ε =( 1 c 1 )>0, b =σ c 1 c 5 >0. (14) Further we assume that ε is a small parameter. Then the right hand side of (13) is a small perturbation. Taking into account the expression for c 1 givenin(9),wecanconcludethat c 1 is a small parameter on condition that the spectral filtering β has asmallvalue,orthemoduleoflineargainorlosscoefficient δ hasalargevalue,orwhenbothconditionsarefulfilled. 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 c 3 = 5/2.Ifwechooseanothervaluefor c 3, 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.

6 6 Computational Methods in Physics Taking into account the value of c 3,(13)canbewrittenas the following system: X=Y, Y=X ( 5 2 )X3 +X 5 + ε( 1+ bx 2 )Y, (15) where the dot means differentiation with respect to τ. The perturbation functions in this system are p (X, Y) =0, q(x, Y) =( 1+ bx 2 )Y. (16) The unperturbed system (i.e., system (15)with ε =0)is X=Y, Y=X ( 5 2 )X3 +X 5. The last system is Hamiltonian with Hamiltonian function (17) H (X, Y) =( 1 2 )Y2 ( 1 2 )X2 +( 5 8 )X4 ( 1 6 )X6 =h, (18) where the parameter h corresponds to a constant Hamiltonian level. System (17) has the following equilibrium points, (0, 0), which is a saddle point, (±X 2,0) = (±1/ 2, 0), whichare centers, and (±X 4,0)=(± 2, 0), which are saddle points. The phase portrait of system (17) is symmetrical concerning X-andY-axis and is shown in Figure 3. Note that this phase portraitcorrespondsto Subcase 6.1 given in Appendix A. 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. Becauseofthis,wewillfurtherbeinterestedonlyinthefamily of closed trajectories localized within the right half of the figure eight loop, that is, the closed trajectories localized between the center (X 2, 0) = (1/ 2, 0) and the homoclinic loop. Each phase trajectory corresponds to a constant value of the Hamiltonian function; that is, H(X, Y) = h, h=const. The value of the Hamiltonian function on the homoclinic trajectory is h hom = 0 andinthe center thevalueis h c = 11/96. It can be easily calculated that the homoclinic trajectory intersects the positive part of the X-axis at the point X hom = (15 33)/8 = Agivenclosedtrajectory localized within the right half of the figure eight loop has the following equation: Γ 0 (h) :( 1 2 )Y2 ( 1 2 )X2 +( 5 8 )X4 ( 1 6 )X6 =h, ( 11 (19) 96 )<h<0. Having these notions in mind, we can obtain the Melnikov function for the system (15): M (h) = (q (X, Y) dx p(x, Y) dy) Γ 0 (h) = ( 1 + bx 2 )YdX. Γ 0 (h) (20) v 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. Introducing the functions u I k (h) = X 2k Y dx, k=0,1, (21) Γ 0 (h) P (h) = I 1 (h) I 0 (h), (22) the Melnikov function and its derivative with respect to h can be written as M (h) = I 0 (h) + bi 1 (h) =I 0 (h) [ 1 + bp (h)], (23) M (h) = I 0 (h) [ 1 + bp (h)]+ bi 0 (h) P (h). (24) Note that I 0 (h) and I 1 (h) are Abelian integrals. We will look for the zeros of M(h) with ( 11/96) < h < 0. To find these zeros we need additional investigations of the functions I 0 (h), I 1 (h), andp(h). Thefactsnecessaryforthis purpose are collected in Lemma B.1 presented in Appendix B. A crucial factor in this consideration is the fact that the function P(h) is positive and strictly monotone decreasing in the interval ( 11/96) < h < 0.ThegraphofthefunctionP(h) obtained by numerical integration is shown in Figure 4. AgivenzerooftheMelnikovfunctionh 0 satisfies the equations M(h 0 )=0, 1+ bp (h 0 )=0. (25) Now we can make some conclusions. Since P(h) is a strongly monotone decreasing function, the second equation in (25) can have only one zero. The function P(h) satisfies the inequality < P(h) < 0.5. Therefore the Melnikov

7 Computational Methods in Physics 7 P(h) h 0.00 v u Figure 4: The graph of the function P(h) = I 1 (h)/i 0 (h) obtained by numerical integration of the Abelian integrals given by (B.7) with the help of Mathematica. 0.2 function has a single zero in the case when the following inequality is satisfied: 2< b < (26) In this case the system (15) hasalimitcycleγ ε (h 0 ),whichis localized in O( ε)-neighborhood of the closed curve Γ 0 (h 0 ). According to the Melnikov theory, the stability of the limit cycle is determined by the sign of quantity εm (h 0 ):stablefor εm (h 0 )<0and unstable for εm (h 0 )>0[27]. From (24) and LemmaB.1 in Appendix B,therefollows that M (h 0 )= bi 0 (h 0 )P (h 0 )<0. (27) Taking into account the inequality ε >0,weconcludethat the limit cycle Γ ε (h 0 ) is stable. Figure 5 shows the limit cycle of system (15)obtainedfor h 0 = and b = Figure 5 presents the results from the calculation of three different initial conditions for the time τ = One of the initial conditions (0.865, 0) coincides with the limit cycle and the corresponding phase trajectory stays the same during the calculation. Another initial condition (0.815, 0) is inside the limit cycle and with the increase of time the trajectory approaches the limit cycle from the inside. The last initial condition (0.915, 0) 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 thetype(15), having in advance assigned limit cycle. Let h= h 0 be the Hamiltonian level around which we want to a limit cycle to arise. After computing P(h 0 ),wegetfrom(25) b = 1 P(h 0 ), ( )<h 0 <0. (28) The obtained in this way value of b provides system (15)with the desired limit cycle. 0.3 Figure5:Thephaseportraitofsystem(15) obtained by numerical solution. 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) canhavealimitcyclethatarisesina 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.onaccountofthis,thelimitcyclesinthetwocaseshave different properties and this leads to different behaviors of the initial wave equation. Theprocessofarisingoflimitcyclesinaperturbed 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 wasdevelopedbyroussarieandshowninaformconvenient for our applications in [43]. Useful and detailed information on these subjects is given in the book of Han and Yu [44]. Formallytheanalysisbelowwillbeperformedforthecase c 1 < 0, c 3 > 0,andc 5 = 0.(Notehowever,thatc 1, c 3 in this paragraph are short notations of the magnitudes c 1, c 3 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].

8 8 Computational Methods in Physics After applying the scaling (10)takestheform where d 2 X dτ 2 u= c 1 c 3 X, ξ = 1 c 1 τ, (29) X+X3 = ε ( 1 + bx 2 ) dx dτ, (30) ε =( 1 c )>0, b = σ( 1 )>0. (31) c 1 c 3 Further, we will assume that ε is a small parameter. (Having in mind the definition of c 1 = 2Kin Appendix C this requires that the parameter K be sufficiently large.) Equation (30) can be written as a perturbed Hamiltonian system in the following way: X=Y+ ε[ X+( 1 3 ) bx 3 ], Y=X X 3, (32) where the dot means differentiation with respect to τ. The perturbation functions in this system are p (X, Y) X+( 1 3 ) bx 3, q(x, Y) 0. (33) The unperturbed system (with ε =0) is Hamiltonian with Hamiltonian function H (X, Y) =( 1 2 )Y2 ( 1 2 )X2 +( 1 4 )X4 =h. (34) Moreover,theunperturbedsystemhasthreeequilibrium points a hyperbolic saddle point (X, Y) = (0, 0) and two nondegenerate centers (X, Y) = (±1, 0). Thevalueofthe Hamiltonian function at the two centers is H(X, Y) = h = 1/4. It is necessary to note that the point (0, 0) is also an equilibrium point saddle point for the perturbed system (32). The equation H(X, Y) = h = 0 defines a symmetric double homoclinic loop L 0 (figure eight-loop), consisting of two homoclinic orbits L + 0 and L 0 connecting the saddle point (0, 0) to itself; that is, L 0 L 0 L+ 0.Thehomoclinic orbits L + 0 and L 0 are localized, respectively, in the half planes X>0and X<0. The solution of the unperturbed system along the homoclinic orbits L + 0 and L 0 can be expressed in thetimedomaininthefollowingway: X=φ 0 (τ) =± 2 sech τ, Y=ψ 0 (τ) = 2 sech τ th τ. (35a) (35b) On theother hand, theequation H(x, y) = h defines three continuous families of periodic orbits {Γ (h) : ( 1/4) < h < 0, X < 0}, {Γ + (h):( 1/4)<h<0,X>0},and{Γ(h) : h > 0}. The periodic orbits Γ (h) and Γ + (h) surround the centers (±1, 0) andtend,respectively,tol 0 and L+ 0 as h (0 ).The limit of Γ(h) as h (0+)isthe double homoclinic loop L 0. In general, the Melnikov function can be defined on each of the mentioned three families of periodic orbits and we have threemelnikovfunctions[44]. Let us introduce the Melnikov function M(h) for the perturbed Hamiltonian system (32) defined along the periodic orbit Γ + (h). Itiswellknown[43, 45] that the Melnikov function has the following asymptotic expansion near h=0(0 < h 1,orh (0 )) where M (h) =s 0 +s 1 h ln h +s 2 h+s 3 h 2 ln h +s 4 h 2 +, (36a) s 0 =M(0), s 1 =( p X + q Y ). (36b) (X,Y) saddle point The quantity M(0) is the Melnikov function computed along the separatrix L + 0 (the computation along L 0 is identical andleadstothesameresults)andforthesystem(32) ithas the form + M (0) = [q (φ 0,ψ 0 ) φ 0 p(φ 0,ψ 0 ) ψ 0 ] dτ. (37) The coefficients s 0, s 1, s 2,...play an important role in the study of the limit cycle bifurcation. According to a theorem of Roussarie [43], if the following relations hold s 0 =s 1 = =s k 1 =0, s k =0, (38) then for small ε = 0 the system (32) canhavethegreatest number of limit cycles k in a small neighborhood of the homoclinic loop. In our case it is easy to compute s 1 = 1=0.Thusthe equation s 0 = M(0) = 0 gives the condition for which in system (32) asinglelimitcyclebifurcatesfrom thehomoclinic loop of the unperturbed system. Taking into account (32), (33), (35a), (35b), and (37) and having conducted some straightforward calculations, we obtain Hence + M (0) = [ p (φ 0,ψ 0 ) ψ 0 ] dτ + = [φ 0 ( 1 3 ) bφ 3 0 ](φ 0 φ 3 0 ) dτ =( 1 45 )( b) = 0. (39) b = σ( c 1 c 3 )= 5 4. (40) Equation (40) expresses the condition for arising of two homoclinic orbits L + ε and L ε connecting the saddle point (0, 0) to itself in system (32). It can be said, that under conditions (40), the perturbed Hamiltonian system (32)

9 Computational Methods in Physics 9 admits a single homoclinic limit cycle L ε,consistingofthe orbits L + ε and L ε ;thatis,l ε L ε L + ε. Let us remember that the perturbations in system (32)preservetheequilibrium point (0, 0) unchanged. The limit cycle L ε is an isolated homoclinic loop located in a small neighborhood of the loop L 0 and passing through the point (0, 0). Further, the stability of the limit cycle L ε is determined by thesignofthesaddlepointquantity[46] σ 0 ={ X (Y + ε[ X+(1 3 )X3 ]) + Y (X X3 )} (X,Y)=(0,0) = ε <0. (41) This means that the separatrix cycles L + ε and L ε,aswellasthe limit cycle L ε,arestable. Having in mind the derivation of (C.6) in Appendix C it is clear that the obtained here (40) precisely coincides with condition (C.2) 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 (C.3). (Note that in(35a), (35b) a 0 = 2). 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 (C.4)) in fact coincides with the same relation in (C.3). Consequently, the numerical confirmation of (40)and therefore of (C.3) coincides with the numerical confirmation of (C.4). 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 CCQ- GLE 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 c j, 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.Atthispoint we do not have any numerical confirmation of the results presented in Section 4. Wehavefixedthevalueof c 3 =5/2, assuming, that c 1 = 1, c 3 =5/2, c 5 = 1.Asaresult,we have obtained the following values of the parameters of (1): δ = , β = , ε = , μ = , ] = The application of the Melnikov method leads to additional condition for b given by (26). What we can say at this point is that the parameters β, ε, μ, ] are in regions for which stable localized fixed-shape solutions and localized pulsating solutions of (1)areidentifiedin[14], while the values of δ and γ = are larger than the usual values [14, 16]. In order to confirm the limit cycles predicted by our approach in Section 4, weplanafurthernumerical 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 (C.2) 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 (C.3) as well as (17) in[30]). As has been mentioned in Section5,thereappearsaprobleminthe numerical confirmation of (C.3) and (C.4). To prove the predicted relation between the critical valueofvelocityandtheamplitudeofthesolitarywave solution (C.3) or of the equilibrium values of the amplitude andvelocityofperturbedsolitonsolution(c.4), a detailed numerical investigation has been performed in [31]. Equation (1) with] = μ = ε = 0 has been numerically solved by means of the split-step Fourier method, applying the Blow- Wood RK4 scheme for the following values of parameters δ ( ), β ( ),andγ ( ) [31]. The obtained results in [31] confirmwithanexcellent accuracy the relation between the equilibrium values of the amplitude and velocity of perturbed soliton solution (C.4) and therefore of (C.3) 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 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 equationfortheamplitudeofthenonlinearwave.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

10 10 Computational Methods in Physics 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: u +c 1 u+c 3 u 3 +c 5 u 5 =0, c 3 >0. (A.1) This equation is equivalent to the system u=v, V = c 1 u c 3 u 3 c 5 u 5. (A.2) Under the condition c 3 >0, we will receive the equilibrium points for the system (A.2) and their type in the case where the system is nondegenerate. In general, equilibrium points are given by the following expressions: (u, V )=(0, 0), (u, V )=(± ( c 3 ± c3 2 4c 1c 5 ),0). (2c 5 ) (A.3) 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 (A.2). Having this in mind, we can make the following classification of the equilibrium points of system (A.2) depending on the coefficients. Case 1 (c 2 3 4c 1c 5 <0, c 1 <0, c 5 <0). System (A.2) has a single equilibrium point (0, 0), which is a saddle point. Case 2 (c 2 3 4c 1c 5 <0, c 1 >0, c 5 >0). System (A.2) has a single equilibrium point (0, 0),whichisa center. Notethatfrom the inequality c 2 3 4c 1c 5 <0there follows that the coefficients c 1 and c 5 have the same signs. Case 3 (c 2 3 4c 1c 5 > 0, c 1 > 0, c 5 > 0). System (A.2) has a single equilibrium point (0, 0),which is a center. Case 4 (c 2 3 4c 1c 5 >0, c 1 >0, c 5 <0). System (A.2) has 3 equilibrium points which are (±u 2, V 2 ) (u 1, V 1 )=(0, 0) -to center, =(± ( c 3 c3 2 4c 1c 5 ),0)-to saddle points. (2c 5 ) (A.4) Case 5 (c 2 3 4c 1c 5 >0, c 1 <0, c 5 >0). System (A.2) has 3 equilibrium points which are (u 1, V 1 )=(0, 0) -to saddle point, (±u 2, V 2 )=(± ( c 3 + c3 2 4c 1c 5 ),0)-to centers. (2c 5 ) (A.5) Case 6 (c 2 3 4c 1c 5 >0, c 1 <0, c 5 <0). System (A.2) has 5 equilibrium points which are (u 1, V 1 )=(0, 0) -to saddle point, (±u 2, V 2 )=(± ( c 3 + c3 2 4c 1c 5 ),0)-to centers. (2c 5 ) (±u 4, V 4 ) =(± ( c 3 c3 2 4c 1c 5 ),0)-to saddle points. (2c 5 ) (A.6) 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: c 2 3 > 16c 1c 5 /3 > 4c 1 c 5, c 1 <0, c 5 <0. Subcase 6.2: 16c 1 c 5 /3 > c 2 3 >4c 1c 5, c 1 <0, c 5 <0. Subcase 6.3: c 2 3 = 16c 1c 5 /3 > 4c 1 c 5, c 1 <0, c 5 <0. B. Analysis of the Auxiliary Functions I 0 (h), I 1 (h),andp(h) The basic properties of the functions I 0 (h), I 1 (h),andp(h) are collected in the following Lemma.

11 Computational Methods in Physics 11 Lemma B.1. The following statements are valid: (a) (b) (c) I 0 (0) =( ) [(11 3) 128 (d) I 1 (0) =( ) [(165 3) 1024 (e) (f) I 0 (h) >0, I 1 (h) >0; (B.1) I 0 (h) >0; (B.2) ] ln [ ( ) 11 ] ln [ ( ) 11 ] ; (B.3) ] ; (B.4) P(h c )=P( 11 ) = 0.5; (B.5) 96 P (0) = I 1 (0) I 0 (0) ; (B.6) (g) the function P(h) is positive and strictly monotone decreasing in the interval ( 11/96) < h < 0, whereupon its derivative is negative; that is, P (h) < 0 for ( 11/96) < h < 0, and therefore P(h c ) > P(h) > P(0) > 0. 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 I 0 (h) = T 0 (h) > 0, wheret 0 (h) is the period (with respect to time) of the closed trajectory Γ 0 (h). Statements(c),(d),and(f) are proved by direct calculation of integrals I k (0) (with h= 0), k = 0,1. Finally, statement (g) for monotonicity of the function P(h) is proved by using the method of Li and Zhang given in [49]. We have calculated the function P(h) numerically. From the Hamiltonian function there follows that Y= ( 1 3 )X6 ( 5 4 )X4 +X 2 +2h. ThentheAbelianintegralsaregivenby I 0 (h) =2 X 3 (h) X 2 (h) ( 1 3 )X6 ( 5 4 )X4 +X 2 +2hdX, (B.7) X 3 (h) I 1 (h) =2 X 2 ( 1 X 2 (h) 3 )X6 ( 5 4 )X4 +X 2 +2hdX. (B.8) The boundaries of integration X 2 (h) and X 3 (h) represent the crossing point of the curve Γ 0 (h) with the axis X. Forthe values of h in the interval 11/96 h 0 we have calculated the following: (1) the angle α, α=α(h) = Arccos ( (5 192h) ); (B.9) 27 (2) the boundaries of integration in the integrals (B.7): X 2 (h) = ( 3 2 ) cos [(α 3 ) ]+( 5 4 ), X 3 (h) = ( 3 2 ) cos [(α 3 ) 1200 ]+( 5 4 ); (3) integrals I 0 (h)ki 1 (h) given by (B.8); (4) the function P(h) = I 1 (h)/i 0 (h). C. Earlier Application of Duffing-Van der PolEquationinDescriptionof the Perturbed Soliton Solution (B.10) 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]): u + c 1 u+ c 3 u 3 =ε( μ μ 1 u 2 )u, (C.1) where c 1 = 2K, c 3 =2, μ = 4βM/γ, μ 1 = 4,andε is a small parameter [30]. K and M have the same meaning as in (2). The amplitude of the perturbed solitary wave solution a 0 is given by a 2 0 = 2 c 1/ c 3 =2K. It has been shown (see (16)in[30]) that if μ C = 2μ 1a 2 0, (C.2) 5 the velocity (frequency) M C and the amplitude a 0 of the perturbed solitary wave solution are related through the relation (see (17)in[30]): M C = 2γa2 0 5β. (C.3) 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 thesamerelationshiphasbeenderivedastheonegiven by (C.3) between the equilibrium soliton amplitude η and velocity k (see (5) in [30]): k = 2γη2 5β. (C.4)

12 12 Computational Methods in Physics Assuming that ε μ = 1and εμ 1 = σ, weget σ =μ 1 / μ. Equation (C.1) canbethenwrittenas u + c 1 u+ c 3 u 3 =( 1+ σu 2 )u. (C.5) Formally (C.5) is precisely the same as (11)withc 5 =0,which is discussed in Section 5,butallcoefficientsarewithahat. By means of quantity σ,thecondition(c.2) can bewritten as σ = 5 (2a 2 0 ) = 5 c 3 (4 c 1 ). (C.6) (C.6) 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, wehaveobtainedthe precise condition (C.2) 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 (C.3) [30]. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. 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