Stability of Dissipative Optical in the 2D Complex Swift-Hohenberg Equation P Yoboue, A Diby, O. Asseu, A. Kamagate

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P Yoboue, A Diby, O. Asseu, A. Kamagate. Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation. International Journal of Physics, 2016, 4, pp.78 - 84. ￿10.12691/ijp-4-4-2￿. ￿hal-01842252￿

HAL Id: hal-01842252 https://hal.archives-ouvertes.fr/hal-01842252 Submitted on 9 Aug 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. International Journal of Physics, 2016, Vol. 4, No. 4, 78-84 Available online at http://pubs.sciepub.com/ijp/4/4/2 © Science and Education Publishing DOI:10.12691/ijp-4-4-2

Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation

P. Yoboue1, A. Diby2, O. Asseu1,3,*, A. Kamagate1

1Ecole Supérieure Africaine des Technologies d’Information et de Communication (ESATIC), Abidjan, Côte d’Ivoire 2Université Félix Houphouët Boigny, Abidjan, Côte d’Ivoire 3Institut National Polytechnique Félix Houphouët Boigny (INP-HB), Yamoussoukro, Côte d’Ivoire *Corresponding author: [email protected] Abstract This article deals with stationary localized solutions of the (2D) two-dimensional complex Swift- Hohenberg equation (CSHE). Our approach is based on the semi-analytical method of collective coordinate approach. According to the parameters of the equation and a suitable choice of ansatz, the stationary dissipative solitons of the 2D CSHE equation are mapped. This approach allows to describe the influence of the parameters of the equation on the various physical parameters of the pulse and their dynamics. Finally, the major impact of spectral filtering terms on the dynamic of the solitons is demonstrated. Keywords: dissipative , spatio-temporal, collective coordinate approach, Ginzburg-Landau equation, complex Swift-Hohenberg equation, spectral filtering Cite This Article: P. Yoboue, A. Diby, O. Asseu, and A. Kamagate, “Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation.” International Journal of Physics, vol. 4, no. 4 (2016): 78-84. doi: 10.12691/ijp-4-4-2.

decade leading to an impressive number of works in several fields of nonlinear science [4,6,7,8]. However 1. Introduction most of these stable soliton solutions studies use purely numerical approaches [9,10]. Solving numerically the Since several years, the stability domain of the spatio- equation for a given set of parameters and a given initial temporal dissipative soliton self-confined in the temporal condition is an extremely lengthy and costly procedure, and spatial dimensions remains a serious issue of which can take up to several days in a standard PC. [1]. The choice of relevant nonlinearity Alternative variational semi-analytical methods can for such self-confinement and ensure a stable solution are overcome this difficulty [11]. These theoretical tools can thorny. Furthermore, the relation between the perceive soliton solutions more efficiently and envisage dimensionality and nonlinear effect used is especially their domains of existence with relative flexibility [12,13,14]. essential. Recently, it has been demonstrated in our previous It should be specified that dissipative systems in works that the collective variable approach [15] is a useful nonlinear optics admit stable solitons in one, two, and tool and reduces significantly the computation time for three dimensions [2]. These solitons can also be purely predicting approximately the domains of existence of temporal, spatial or spatio-temporal. Dissipative soliton stable light bullets in the parameter space of the (3D) has been widely studied in nonlinear dissipative optics, complex cubic-quintic Ginzburg-Landau equation [13]. from fundamental point of view and due to the clear This present work provides evidence for the stationary physical meaning in particular application. Important solutions of the (2D) complex Swift-Hohenberg equation, applications are passively mode-locked laser systems and which, to our knowledge has not been enough reported in optical transmission lines. the literature. This equation is useful in the description of Indeed, the formation of this dissipative structure is the dynamics of dissipative solitons in laser cavity in much more complex than that of conservative systems, experimental situations. because in addition to the right balance between the dispersion and the nonlinearity, dissipative solitons exchange energy and (or) matter with an external source. 2. Materials and Methods They exist only when there is a continuous energy supply to the system. Whenever the energy supply is stopped, 2.1. Theory of dissipative Soliton in soliton “stops living.” Their shape, amplitude, velocity are Swift-Hohenberg System all fixed and defined by the parameters of the system [3] rather than by the initial condition. One of the main properties of wave is that they tend to The properties and conditions of their existence have spread out as they evolve. The principal cause for this is been studied extensively [4,5]. The theoretical study of that distinct frequency; that are superposed to create the these soliton has recently received a boost during the past wave packet, propagate with different velocities and (or) 79 International Journal of Physics in different directions. It knows that generally the D 1 24 nonlinear effects accelerate the spreading of the wave. ψz−−−iii ψ tt ψ rr γψψνψψ − i 22 (1) Nevertheless, under certain conditions, nonlinearity may 24 compensate the linear effects. The resulting balanced =+δψεψψβψµψψγψ ++tt +2 ttt localized pulse or beam of light, which propagates without where , , , , , , and are real constants. Without decay, is generally known as soliton. Therefore, optical the additive term this equation is the same as the solitons are localized waves that propagate stably in 2 CGLE 휇one.훿 훽Th퐷e physical휈 훾 훾 meaning휀 of each term depends nonlinear media with dispersion, diffraction or both. 2 푡푡푡 on the real problem훾 휓which must be examined. In optics, Originally, the terminology soliton was reserved for this equation describes the laser systems [23,24], optical conservative systems and particular set of integrable regeneration for optical fibre transmission systems with solutions existing as a result of the delicate balance soliton signals [25], nonlinear cavities with an external between dispersion (diffraction) and nonlinearity. pump [26] and the parametric Oscillator [27]. When However, similar classes of stable self-sustained applied to the propagation of the pulses in a laser system, structures can be found for a wide range of physical as is the case in our study = ( , , ) represents the systems far from equilibrium. The term normalized optical envelope and is function of three real has been used by Nicolis and Prigogine [16] to describe t 휓 휓 푟 푡 푧 these systems far from equilibrium. As a new paradigm of variables. With the retarded time in the frame moving z nonlinear waves, solitons in real dissipative environments with the pulse, is the propagation distance or the cavity are known as dissipative soliton [4]. The dissipative round-trip number. And finally r r = x + y soliton has different characteristics than those of represents the transverse coordinate, taking account2 of the2 conservative systems. They are attracting a significant spatial diffraction effects. � � � surge of research activities on their spatial (temporal) The left-hand side contains the conservative terms: complexity during the last years, particularly for systems namely, = +1( 1) which is for the anomalous (normal) modelled by the cubic quintic complex Ginzburg-Landau dispersion propagation regime and which represents, if equations [4,5]. Apart from the balance between negative,퐷 the saturation− coefficient of the Kerr nonlinearity. dispersion (diffraction) and nonlinearity, the separate In the following, the dispersion isν anomalous, and is balance between gain and loss is essential for the pattern kept relatively small. The right-hand-side of equation (1) formation of the dissipative soliton. This second balance is includes all dissipative terms: , , and are νthe very crucial, and gives the dissipative soliton a markedly coefficients for linear loss (if negative), nonlinear gain (if different dynamic from that of conservative solitons. The positive), spectral filtering (if positive)훿 휀 훽 and saturation휇 of shape, amplitude and widths of the dissipative soliton are the nonlinear gain (if negative), respectively. And finally fixed, depend drastically on the system parameters [17] represents the higher-order spectral filter term, which is and may evolve stationary, periodically, or even very important in this present study. 2 chaotically [10,18,19]. One of the generic equations 훾 describing the dynamics of dissipative solitons and that we have intensely studied is the complex Cubic-quintic Ginzburg-Landau equation model [4,20,21]. This equation could be applied to the modeling of a wide-aperture laser cavity with a saturable absorber in the short pulse regime of operation. The model includes the effects of two- dimensional transverse diffraction of the beam, longitudinal dispersion of the pulse, and its evolution along the cavity. The spectral filter of this model is restricted to the term of second order and can only describe a spectral response with a single maximum. However, experiences indicate that the gain spectrum is usually wide and can have multiple peaks. To be more realist we need to add others terms of higher-order spectral filtering to the complex Cubic- Figure 1. Evolution of the spectral response in the case of the complex quintic Ginzburg-Landau equation (CGLE), this lead to Cubic-quintic Ginzburg-Landau equation (in circle) and of the complex the complex Swift-Hohenberg equation (CSHE). Swift-Hohenberg equation (in solid curve). The other parameters appear The complex Swift-Hohenberg equation is useful to inside the figure describe soliton propagation in optical systems with linear In order to have stable pulses in the frequency domain, and nonlinear gain and spectral filtering such as communication links with lumped fast saturable absorbers must be positive and can have both sign (positive or or fiber lasers with additive-pulse mode-locking or negative) contrary to the case of the CGLE equation 훾2 훽 nonlinear rotation. It is clear that the higher where must be strictly positive. If is greater than zero, order of the spectral filter is extremely essential to analyze we obtain a spectral response with a single maximum, on the generation of more complex impulse. the other훽 hand ( less than zero)훽 we have now two According to this equation, we will investigate the maxima. To support this comments, the effect of the steady state of the 2D stationary solutions. It describes as spectral filter is shown훽 in Figure 1. It is described by the well quantitatively as qualitatively many nonlinear effects following transfer function: which occur during the propagation and can be read in this 24 normalized form [4,22]: T(ω) = exp ( δ −− βω γ2 ω ). (2) International Journal of Physics 80

The circle curve shows the spectral response in the case By neglecting the residual field ( = 0 ) the bare of the complex Cubic-quintic Ginzburg-Landau equation approximation, as is the case in most practical studies [10], CGLE (with = 0 ), it is a Gaussian curve with we chose a Gaussian function as ansatz푞 function that is amplitude and width . This curve has a single given by the following: 2 maximum. In the훾 case of the complex Swift-Hohenberg 22 equation (CSHE),훿 the response훽 of spectral filter is much tri i f= Aexp  −−+c t22 + c r + ip . (4) affected and depends on three parameters with positive. 22 tr wwtr22 This situation gives a spectral response with two distinct 2 maximums. In this work we will carefully investigateγ this So, here we are assuming that all the pulses are purely situation where is nonzero. Gaussian with spatial and temporal chirp and do not The parameter values are chosen according to that the consider other forms of pulses. 2 width and heightγ of the two spectral responses are not too In such case, the field is necessarily ( , , ) = , with different. However, as the spectral response is different, it , the spatial and temporal variables along and axis goes without saying that the (2D) two-dimensional respectively. , , , , , and휓 푟 represents푡 푧 푓 the dissipative soliton in both case are different profiles for 푡 푟 푡 2 푟2 collective variables. 푡Stands푟 for푡 soliton푟 amplitude, the same value of the cubic gain . 2 2 퐴 푤 푤 푐 푐 푝 and represent the temporal and spatial widths푡 respectively. /(2 ) is the parameter of the chirp√ along푙푛 푤 휀 푟 2.2. Stability Study axis,√ 푙푛/(2 푤) the parameter of the spatial chirp and is 푡 the global phase푐 휋that evolves along with propagation.푡 Our study, really, is to examine the stationary soliton of 푟 the 2D complex Swift-Hohenberg equation by semi- When푐 a stationary휋 regime is reached, the phase becomes푝 a analytical approach, as far as we knew this has not been linear function of the propagation distance . done previously. More specifically, our major goal is to After this choice of the ansatz function, variational provide an approximate mapping of the regions of existence analysis could be carried out by neglecting푧 the residual of stable and unstable solutions in the parameter space of field (the bare approximation). Applying the bare the equation (1), as we have done previously with the 3D approximation to the 2D CSHE, consists in substituting complex Cubic-quintic Ginzburg-Landau equation one [13]. the field by the given trial function ( = ) and To achieve this goal, we use the collective variable projecting the resulting equations in the following direction: approach, which helps to simplify the characterization of 흍 풇 흍 풇 ∂f * the pulse by use of a low dimensional equivalent ( X = Aw,.t, w rtr ,,, c c p) mechanical system based on a finite number of degrees of ∂X freedom. Each degree of freedom can then be described by A jet of six differential equations which govern the means of a coordinate called the collective variable. evolution of the optical pulse parameters propagating in Indeed, the propagation of the pulse describes not only the space and time is obtained: pulse as a collective entity (localized in time and space) but also all other localized or non-localized excitations, 335 25 A =+−+− Aδε A A βµ A Ac D such as noise or radiation, which are always more or less 2 t 49wt present in the real system. For a better understanding of these dynamic processes, it is important to develop 3 −+2Ac 32 c2 −+ w 24 c Aγ , analytical approaches to help to bring the dynamics of the r t tt 4 2 w pulse to that of a simple mechanical system with only a t 12 small number of degree of freedom. w =−−2 wcD wA24εµ A w The mean idea in the collective variable approach is to t tt49 t t associate collective variables with the pulse’s parameters 422β 84 12 of interest for which equations of motion may be derived. +−(11wctt) +( wc tt −) γ 2, w w3 One may introduce collective variables, dependent; t t = 1, 2, . . , 12 say with , in a way such that each of them w =−−4, wc wA24εµ Aw can correctly describe푁 a fundamental parameter푧 of the r rr49 r r 푖 pulse푋 (amplitude,푖 width,푁 chirp …) [15,28]. To this end, 1 81 one can decompose the field ( , , , ) in the following way: c =2 −− cD22 cβγ − A t4 tt 22 wt ww tt2 ψ (rtz,,) = f( x12 ,휓 x ,푥 …+푦 xn푡 , t푧) q( zt ,) (3) 4 1 −+ A4ν 48cc+ 2 γ , where the ansatz function is a function of the collective 2 tt4 2 variables and is chosen to draw, at best, the configuration 9wt wt of the pulse.푓 And ( , ) is a residual field that represents 1 44 =−−22γν +− 4 all other excitations in the system (noise, radiation, dressing ccrr 4, A A 29w2 ww 42 field, etc.) [15]. The푞 choice푧 푡 of the trial function that introduces r rr 3D 25 the collective variables in the theory is important for the p =2βγ cA +24 −−+A ν success of the technique. After choosing the ansatz function t 22 49wwtr one can pursue the process of characterization of the pulse (5) 1 by neglecting the residual field. This approximation is −−12c 2 wc22 γ . called the bare approximation [15]. In this way one can t2 tt 2 wt consider the fact that the pulse propagation can be completely characterized that the ansatz function. 81 International Journal of Physics

It is important to point out that these equations give no 2.3. (2D) CSHE Stationary Dissipative Soliton explicit information with regard to the different solutions of the equation CSHE (1) and their stability. Thereby, they The stationary solutions of the 2D complex Swift- give us the first idea on the dynamic of the light pulse. Hohenberg equation exist in finite regions in the They simply reveal in detail the influence of each equation parameter space. Nevertheless, it would be extremely CSHE (1) parameters on the various physical parameters difficult to map in extensor these regions in the all of the soliton. dimensional parameter in which we operate. For this Thus, one’s can clearly see that spectral filter raison, we restrict ourselves to fix all the parameters coefficients ( and ) affect the amplitude of the pulse, except for two which vary. By setting key parameters and ( ) its temporal width, temporal chirp and the global phase. by individually varying the nonlinear gain and the 2 However, these훽 parameters훾 have no formal effect on the Kerr term saturation of the optical nonlinearity ( ), we 휀 spatial variables. The temporal ( ) and spatial widths ( ) have revealed in the ( , ) plane, the stationary solution of ν also depend on the nonlinear gain ( ) and its saturation ( ). the 2D complex Swift-Hohenberg equation. By 푡 푟 휈 휀 As expected, the terms of spectral푤 filtering coefficients푤 ( investigating the parameter regions situated in the = = 1 = 0.3 and ) and dispersion term ( ) affect휀 the temporal width휇 neighbourhood of the parameters , , = 0.5 = 0.1 = 0.05 and have no action on the radial component. 훽 , and , with an initial pulse 2 퐷 훾 훽 − Similarly,훾 the spatial and퐷 temporal chirp parameters 훿 − 휇 − 훾2 tr22 are influenced in the same way by the Kerr term saturation ψ (r, t ,0) = 2.86 exp  −− (7) 푟 푡  of the optical nonlinearity푐 ( ), but the푐 temporal term is 0.7 1.36 also affected by the terms of spectral filtering coefficients and by individually varying from -0.3 to -0.16 and ( and ) and dispersion νterm ( ) . Finally, not any from 0.49 to 0.57, so for each value pair ( , ) the parameters of the soliton are influenced by ( ), the global 2 Newton-Raphson allows to look흂 for the fixed point and we휺 phase,훽 but훾 are governed by the second퐷 order spectral filter study its stability. Thus one’s can easily realize흂 휺 the term ( ). 푝 cartography of the solution of the equation (1). Figure 1 In this way, the equation of propagation of the optical 2 shows the mapping of the solutions for the range of wave is훾 transformed into a system of differential equations, selected values. The dotted lines correspond to the fixed describing the evolution of the physical parameters of the points which represent the stationary solution of the 2D pulse (amplitude, width...) during the propagation. complex Swift-Hohenberg equation. It can be noticed that This approach provides the basic parameters of the this stability area is very narrow with low value of the fixed points, and a mapping of different types of solutions, nonlinear gain( ), therefore highly sensitive. That implies thereby reducing by several orders of magnitude the that a small change of can have a real impact on the volume of calculation required usually. system. Thus, 휺all the solitons parameters (amplitude, The fixed points (FPs) of the system are found by width, chirp...) stay 휺stationary throughout propagation. imposing the left-hand side of equation (5) to be zero The total energy of the system also has the same dynamic ( = 0 with = , , , , , ). The threshold of which does not change at all, during propagation for these existence of FPs can be estimated by the relation 푋̇ 푋 퐴 푤푡 푤푟 푐푡 푐푟 푝 values considered. Besides the stationary domain, we have 2 . If > , we have in general both stable and푠 instable fixed points which can be dived in two categories: unstable fixed points. 휀 ≈ 푠 the limit-cycle attractor and the instable solutions. Here, �The훿휇 stability휀 휀of FPs is determined by the analysis of the our main interest is to study the dynamic of the pulse in eigenvalues ( = , , , , , ) of the matrix the dotted line domain. = / 푗 . 푡 푟 푡 푟 The stability휆 criterion푗 퐴 푤 is as푤 follows:푐 푐 if 푝the real part of at 푖푗 푖 푗 least푀 one휕푥 ̇of휕푥 the eigenvalues is positive, the corresponding FP is unstable. Hence, to have stable FP, the real parts of all the eigenvalues of the matrix must be negative. The stable fixed points correspond to stationary 푖푗 solutions of the 2D complex Swift푀-Hohenberg equation (1). In addition, the fundamental parameter which helps to control the state of the solution and study its stability is the total energy Q given by the following equation: +∞+∞ 2 Q( z) = ∫∫2πψ r( r ,, t z) drdt ⋅ (6) −∞ 0

For the dissipative system, the total energy gives us the main information about the soliton dynamics. It’s not Figure 2. cartography of the solutions of the 2D complex Swift- conserved but evolves in accordance with the so-called Hohenberg equation in the ( , ) plane. The stable fixed points regions balance equation. When a stationary solution is reached, in dotted lines represents the domain of stationary solitons of the equation. Other CSHE parameters흂 휺 appear inside the figure the total energy converges to a constant value. However, the soliton, is a pulsating one, the total energy is an This first analysis shows that the 2D complex Swift- oscillating function of z. And finally, when we have Hohenberg equation stationary solutions exist in the space unstable solutions, energy tends to infinity. of selected parameters. In this way, it is more revealing to map the area of stability in space of the spectral filters International Journal of Physics 82 namely in the ( , ) plane. For this, the star point (see Figure 2) in the stability domain of the Figure 2 2 corresponding to훽 훾= 0.22 and = 0.524 is considered. From this point, the stable solutions in ( , ) plane for the selected values휈 , −, , , and휀 are maped. 2 The Figure 3 summarizes this result. The훽 훾 dotted lines represent the stationary휈 훿 훾solutions.퐷 휇 As휀 will be seen from this cartography, the stationary solutions of the 2D complex Swift-Hohenberg equation exist for both positive and negative values of ; the same holds true for . This stability domain is wide with sensitive value of . 2 This first interesting results훾 illustrate that the stationary훽 2 dissipative solutions of the 2D complex Swift-훾Hohenberg equation can be found for any signs of the parameters and . Based on those results, the dynamics of the dissipative soliton depending on the signs of spectral filter훽 2 Figure 3. cartography of the solutions of the 2D complex Swift- and its훾 high-order term (same signs or opposite signs) are Hohenberg equation in the ( , ) plane. The stable fixed points regions carefully examined. in dotted lines represents the ퟐdomain of stationary solitons of the Accordingly, four points marked with a star, a circle, a equation. Other CSHE parameters휷 휸 appear inside the figure square and a triangle are chosen (Figure 3).

Figure 4. (up) evolution of the total pulse energy of the stationary dissipative soliton, and (below) the spectral filter response in case < 0 and > 0

The star in the figure corresponds to the case where When the spectral filtering and its high-휷order term휸ퟐ are these parameters are of opposite signs [ = 0.3 (negative) of the same negative signs [ = 0.3 (negative) and and = 0.05 (positive)]. The characteristics of such a = 0.05 (positive)] represented by a circle in Figure 3, pulse are represented by the Figure훽 4.− The transfer the transfer function also has two훽 maxima− at its ends but 2 2 function훾 of the spectral filtering of that stationary soliton with훾 no− central pulse. So these solutions do not have the with positive and negative has two maxima. The same profile and features as in the situation described total energy of that soliton after a short oscillation remains previously. The solution is still stationary as shown by the 2 constant훽 over long distances.훾 This dynamic characterizes a evolution of the total energy, but has less energy than the stationary solution. previous one due to the value of . This situation is well summarized in Figure 5. 훾2

Figure 5. (up) evolution of the total pulse energy of the stationary dissipative soliton, and (below) the spectral filter response in case < 0 and < 0

In case where the spectral filtering and its high-order The last case studied corresponds to the 휷scenario 휸whereퟐ term are of the same positive signs [ = 0.05 (positive) the spectral filtering and its high-order term are of and = 0.05 (positive)] represented by a square in opposite signs [ = 0.02 (positive) and = 0.03 Figure 3 the transfer function has only 훽a single maximum. (negative)] represented by a triangle in Figure 3. Here, the 2 2 The soliton훾 dynamic is not changed; it is still stationary transfer function 훽of the spectral filtering of훾 the −pulse but has energy much lower than that of the first two cases shown in Figure 7 has the same behaviour as that of the treated. The Figure 6 draws these behaviours. 83 International Journal of Physics circle. The optical soliton has the same dynamic as the previous case, but with even lower energy.

Figure 6. (up) evolution of the total pulse energy of the stationary dissipative soliton, and (below) the spectral filter response in case > 0 and > 0

휷 휸ퟐ

Figure 7. (up) evolution of the total pulse energy of the stationary dissipative soliton, and (below) the spectral filter response in case > 0 and < 0

It appears from the different scenarios studied that the The collective variable approach is very휷 efficient휸ퟐ to spectral filtering and its high-order term parameters ( obtain stable stationary solutions when a suitable trial and ) have a great influence on the dynamics of the function is chosen. This technique is incomparably spectral response. It is certainly true that if stationary훽 quicker than direct numerical computations. 2 solutions훾 regardless the signs of and are obtained, This work can be extensive and we are confident that they have a real impact on the spectral response of the these applications will numerous in the fields of physics, 2 pulse. 훽 훾 chemistry and biology as described by the complex Swift- So when the high-order term is greater than zero Hohenberg equation. ( > 0) and no matter the sign of the spectral filtering, the spectral response has one or two distinct maxima. 2 훾However, in the scenario where the high-order term is References less than zero ( < 0) the spectral response has the same behavior with zero central pulse for any signs of . [1] Y. Silberberg, Collapse of optical pulses, Optics Letters 15 (1990), 훾2 1282-1284. These results clearly show that in the cases of [2] N. N. Rosanov, Spatial hysteresis and optical patterns, Springer experimental; the choices of the spectral filtering훽 and its Berlin, 2002. high-order term ( and ) are very crucial according to [3] N. Akhmediev and A. Ankiewicz, Solitons Around Us: Integrable, the shape of the spectrum. Hamiltonian and Dissipative Systems, Lecture Notes in Physics, 훽 훾2 Springer, Heidelberg, 2003. [4] N. Akhmediev and A. Ankiewicz, Dissipative solitons. Ed. Springer, Heidelberg, 2005. 3. Conclusion [5] N. Akhmediev and A. Ankiewicz, Dissipative solitons: from optics to biology and medicine, Springer, Heidelberg, 2008. At the end of our study, we have demonstrated, for the [6] Q. Zhou, Y. Zhong, M. Mirzazadeh, A.H. Bhrawy, E. Zerrad and first time to our knowledge, the stationary dissipative A. Biswas, Thirring combo solitions with cubic nonlinearity and spatio-temporal dispersion, Waves in Random and Complex solutions of the 2D complex Swift-Hohenberg equation Media. Vol. 26 (2016), Issue 2, 204-210. with our collective variable approach. Particularly, the [7] Q. Zhou, M. Mirzazadeh, E. Zerrad, A. Biswas and M. Belic, regions of existence of stationary dissipative soliton in the Bright, dark and singular solitons in optical fibers with spatio- ( , ) and ( , ) planes are shown. It has been also temporal dispersion and spatially-dependent coefficients, Journal shown that the validity of these studies is based on a of Modern Optics. Vol. 63 (2016), Number 10, 950-954. 휈 휀 훽 훾2 [8] M. Mirzazadeh, M. Eslami, M. Savescu, A. H. Bhrawy, A. A. careful selection of the ansatz function. Alshaery, E. M. Hilal and A. Biswas, Optical solitons in Our results reveal the essential character of the spectral dwdmsystem with spatio-temporal dispersion, Journal of filtering and its high-order term parameters ( and ). Nonlinear Optical Physics and Materials. Vol. 24 (2015), Issue 1, Our study shows that it is possible to observe the 1550006. 훽 훾2 [9] J.M. Soto Crespo, Ph. Grelu and N. Akhmediev, Optical bullets stationary dissipative solutions of the 2D complex Swift- and “rockets” in nonlinear dissipative systems and their Hohenberg equation, whatever the signs of and . It transformations and interactions, Optics Express 14 (2006), 4013- has been also shown that theses parameters have a real 4025. 2 impact on the spectral response. 훽 훾 International Journal of Physics 84

[10] J.M. Soto Crespo, N. Akhmediev and Ph. Grelu, Spatiotemporal [19] N. Akhmediev and V.V. Afanasjev, Novel arbitrary-amplitude optical solitons in nonlinear dissipative media: From stationary soliton solutions of cubic-quintic complex Ginzburg-Landau light bullets to pulsating complexes, CHAOS 17 (2007), 037112. equation, Phys. Rev. Lett., 75 (1995), 2320-2323. [11] A. H. Arnous, M. Mirzazadeh, S. Moshokoa, S. Medhekar, Q. [20] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence Zhou, M. F. Mahmood, A. Biswas and M. Belic, Solitons in Dissipative Solitons, Springer-Verglag, Berlin, 1984. optical metamaterials with trial solution approach and backlund [21] C. Normand, Y. Pomeau, M.G. Velarde, Convective instability: A transform of riccati equation, Journal and Computational and physicist’s approach, Rev. Mod. Phys., 49 (1977), 581-624. Theoretical Nanoscience. Vol. 12 (2015), Number 12, 5940-5948. [22] N. Akhmediev and A. Ankiewicz, Solitons of the Complex [12] V. Skarla and N. Aleksic, Stability Criterion for Dissipative Ginzburg-Landau equation, Springer, Berlin, 2001. Soliton Solutions of the One-, Two-, and Three-Dimensional [23] P.A. Belanger, Coupled-cavity mode-locking: a nonlinear model, J. Complex Cubic-Quintic Ginzburg-Landau Equations, Phys. Rev. Opt. Soc. Am. B, 8 (1991), 2077-2082. Lett. 96 (2006), 013903. [24] J.D. Moores, On the Ginzburg-Landau laser mode-locking model [13] A. Kamagaté, Ph. Grelu, P. Tchofo-Dinda, J.M. Soto-Crespo, N. with fifth-oder saturable absorber term, Opt. commun., 96 (1993), Akhmediev, Stationary and pulsating dissipative light bullets from 65-70. a collective variable approach. Phys. Rev. E, 2009, 79, 026609. [25] L.F. Mollenauer, J.P. Gordon and S.G. Evangelides, The sliding- [14] M. Mirzazadeh, M. F. Mahmood, F. B. Majid, A. Biswas and M. frequency guiding: an improved from soliton jitter control, Opt. Belic, Optical solitons in birefringent fibers with riccati equation Lett., 1992, vol.17 (1992), 1575-1577. method, Optoelectronics and Advanced Materials - Rapid [26] W.J. Firth and A.J. Scroggie, Optical bullet holes: Robust Communications. Vol. 9 (2015), Numbers7-8, 1032-1036. controllable localized states of a nonlinear cavity, Phys. Rev. Lett., [15] P. Tchofo-Dinda, A.B. Moubissi, K. Nakkeeran, Collective 76 (1996), 1623-1626. variable theory for optical solitons in fibers. Phys Rev. E, 64 [27] P.S. Jian, W.E. Torruellas, M. Haelterman, U. Peschel and F. (2001), 016608. Lederer, Solitons of singly resonant optical parametric oscillators, [16] G. Nicolis and I. Prigogine, Self-organization in nonequilibrium Opt. Lett, 24 (1994), 400-402. systems - From dissipative structures to order through fluctuations, [28] M. Mirzazadeh, A. H. Arnous, M. F. Mahmood, E. Zerrad and A. John Wiley \& sons, New York,1977. Biswas, Solitons solutions to resonant nonlinear Schrodinger’s [17] W.V. Saarlos and P.C. Hhenberg, Fronts, pulse, sources and sinks equation with time dependent coefficients by trial solution in generalized Ginzburg-Landau. Physica D, 56 (1992), 303-367. approach, Nonlinear Dynamics. Vol. 81 (2015), Issue 1-2. 277- [18] N. Bekki and K. Nozaki, Formations of spatial patterns and holes 282. in the generalized Ginzburg-Landau equation, Phys. Lett., 110 (1985), 133-135.