Dissipative solitons with a Lagrangian approach

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1 Optical Fiber Technology Invited paper Dissipative solitons with a Lagrangian approach Adrian Ankiewicz, Nail Akhmediev, Natasha Devine Optical Sciences Group, Research School of Physical Sciences Engineering, The Australian National University, Canberra ACT 2, Australia Received 16 August 26; revised 31 October 26 Available online 12 January 27 Abstract We apply Lagrangian techniques to dissipative systems described by the complex Ginzburg Lau equation CGLE that is used for modeling passively mode-locked fiber lasers all-optical soliton transmission lines. In particular, using Lagrangian equations, we re-derive the known exact solutions of the CGLE. We also apply the technique to finding approximate solutions for pulsating solitons. 26 Elsevier Inc. All rights reserved. Keywords: Dissipative solitons; Fiber lasers; Lagrangian approach 1. Introduction The dissipative soliton is a new concept that has been developed quite recently 1]. It includes ideas from three major sources, viz. stard soliton theory developed since the 196s, principles from nonlinear dynamics theory Prigogine s ideas of systems far from equilibrium. Physically speaking, the major part of stard soliton theory is the notion of the balance between dispersion nonlinearity that allows stationary localized solutions to exist. However, for dissipative systems, we need to acknowledge that the major balance is between gain loss which is necessary for the solitons to be stationary objects. Nonlinear dynamics inspires us with the idea of soliton bifurcations the chaotic evolution of solitons. Finally, the theory of systems far from equilibrium tells us that solitons are self-organized formations requiring a continuous supply of energy. As soon as that supply finishes, a dissipative soliton ceases to exist. These ideas can be applied to various fiber optic devices such as passively mode-locked fiber lasers 2,3] high-density optical transmission lines 4]. A Lagrangian approach helps us in obtaining a deeper understing of how these systems work in solving particular problems related to fiber optic devices. This article provides a simple introduction to the Lagrangian approach applied to dissipative systems in general to dissipative solitons in particular. The treat- * Corresponding author. address: nna124@rsphysse.anu.edu.au N. Akhmediev. ment is based on the cubic quintic complex Ginzburg Lau equation as it is a master equation for short pulse generation propagation in nonlinear dissipative systems 2 4]. Generally defined, solitons are the simplest localized solutions of a certain class of partial differential evolution equations 5]. First, they were introduced as nonlinear modes of the KdV equation, so that they allow us to solve initial value problems 6], when considered together with radiation waves. Indeed, when a system is integrable, its solutions can be constructed using the inverse scattering technique 5]. Further generalization to non-integrable systems 7] shows that localized traveling wave solutions can be found analyzed using some basic approaches from nonlinear dynamics. One of the powerful techniques that can be applied to these systems is that which employs Hamiltonian equations. These are helpful in formulating the problem in term of a Hamiltonian 8] when the system is integrable, in finding stationary solitons stability regions for them if the problem is not integrable 9]. The notion of solitons can be generalized to cover dissipative systems 1]. Dissipative solitons are already well-defined objects in optics. Various techniques have been applied to investigate them, with numerical simulations being the most powerful. For dissipative systems, there are no conserved quantities that can be used to solve particular problems. In particular, Hamiltonian methods cannot be used here. However, for dissipative systems, we can write down the Lagrangian use Lagrangian equations to derive soliton solutions /$ see front matter 26 Elsevier Inc. All rights reserved. doi:1.116/j.yofte

2 92 A. Ankiewicz et al. / Optical Fiber Technology In this work, we apply Lagrangian principles to dissipative systems described by the complex cubic quintic Ginzburg Lau equation. We formulate Lagrangian equations for the CGLE apply them to re-derive known solutions of this equation, also analyze pulsating solutions that are only known from numerical simulations. In the latter case, solutions can be approximated by various localized trial functions that allow us to reduce the system to a finite-dimensional dynamical system. Solutions in the form of limit cycles for this dynamical system correspond to pulsating dissipative solitons. In the final section, we will present some new results for this topic. The complex Ginzburg Lau equation CGLE is a general model for dissipative systems, describing a vast variety of nonlinear phenomena in physics 11]. In particular, it describes pulse generation by passively mode-locked soliton lasers 12] signal transmission in all-optical communication lines 13]. In dimensionless form, the CGLE is 7,14] iψ z + D 2 ψ xx + ψ 2 ψ = ν ψ 4 ψ + iδψ + iɛ ψ 2 ψ + iβψ xx + iμ ψ 4 ψ. 1 Here the dispersion D =±1, while ν represents the deviation from the Kerr-law response of the fiber. Nonlinear terms ɛ,β,μ,δ are defined in 14]. The application of the variational integral Euler equations has been explained in 15, p. 276], where the use of the Lagrange density its relation to the Euler Lagrange equations for the Schroedinger equation other conservative equations are also given p We adopt the Lagrangian action] L = L d dx, 2 where L d is the Lagrangian density. Here, we have L = i 2 ψ ψ z ψ ψ dx z + D 2 P 1 2 S 4 ν 3 S 6 3 for field ψx,z. Here, we have defined S n = ψ n dx P = ψ 2 x dx. Now, Q = S 2 is the total pulse energy. The rate of change of Q is 14] Q z = dq dz = 2δQ βp + ɛs 4 + μs 6. 4 Now, if we apply the Euler Lagrange equations to the CGLE, we obtain d z ψ z + d x ψx d = R, 5 ψ where a subscript x or z means derivative w.r.t. that variable, * indicates complex conjugate R indicates the dissipative terms of the CGLE, viz., R = i δψ + ɛ ψ 2 ψ + μ ψ 4 ] ψ + βψ xx. 6 For a stationary solution i.e., not a pulsating solution 24], for example, the energy Q is constant so the term Q z = dq/dz is also zero see p. 271 of 14]. This condition constrains the solutions. Of course, if we have a stationary state, then the soliton profile does not depend on z. The solution written as a complex function in general form ψx,z= Bxexp iθz 7 has the complex amplitude function Bx that does not depend on z. Thus ψ = Bx. The phase function θz = ωz is a linear function of z,soθ z = ω. If we use this ansatz, then the first term in the Lagrangian reduces to ω Bx 2 dx = ωq. 8 For the real term P,wehave P = B x 2 dx = B xb x dx, 9 assuming a localized function, i.e., that Bx vanishes at ±. The signs of the parameters in Eq. 1 are defined by the physics of the optical system see 14]. In optics, we usually have δ<, μ<, ɛ>, β>, so then the gain term is ɛs 4. In some cases, we can regard the stationary solution as the one which takes minimum energy from the environment. However, in general it is not sufficient to have a solution constrained by dq/dz = while retaining one free parameter, then maximizing the gain relative to that parameter. Up till now, the Lagrangian has mostly been used to find approximate solutions of conservative systems, like the nonlinear Schroedinger equation NLSE or the NLSE with a non-kerrlaw term i.e., ν see 16 18]. For a trial solution containing parameters f j, j = 1, 2,..., the stard variational approach can be modified to allow for dissipative terms like self-steepening magnetic effects see 19 21]. We have, from Eq. 3, d dz f z f = 2Re R ψ f dx 1 for each parameter f. Our purpose here in this part is quite different in that we seek exact solutions of the CGLE, which is a dissipative system involving a balance between gain loss in various parts of a pulse. Of course, the pulse could be a signal traveling along an optical fiber, an electric or magnetic field distribution, or function giving a distribution of temperature, pressure or number of particles 1]. If we have an ansatz, then the derivative of the Lagrangian w.r.t. any parameter should be equal to an integral involving the

3 A. Ankiewicz et al. / Optical Fiber Technology dissipative terms. This may or may not be zero. We now show how this works. It is thus a new way to find solutions of the CGLE. 2. General formalism for exact solutions In this section, we apply the Lagrangian formalism to find stationary soliton solutions, i.e., solutions which have the form of Eq. 7, of the CGLE. Then, using Eq. 1, we have, for a general parameter f : f d dz f z = 2Re 1 ψ iψ f δ B 2 + ɛ B 4 + μ B 6 + βb xx B ] dx. 11 Now, suppose, the parameter f in Eq. 11 is f = θz. Noting that 1 ψ iψ θ = 1 12 from Eq. 11, we get dq dz = 2Re δ B 2 + ɛ B 4 + μ B 6 + βb xx B ] dx = 2δQ βp + ɛs 4 + μs This result is nothing other than the so-called balance equation 14] for energy Q. Hence, using the phase function as a parameter in the Lagrangian equation verifies Eq. 4. Plainly, the r.h.s. is zero for a stationary solution. The energy of the stationary solution is constant, but this does not indicate the conservation of energy which applies for the NLSE. We recall that, for the NLSE, each conserved quantity is related to a symmetry of the equation Noether s theorem, that energy conservation is related to a phase shift symmetry p. 29 of 14]. In a somewhat analogous manner, for the CGLE here, it is the phase term parameter that results in the energy being constant for the stationary solution. In contrast to conservative cases, stationary solutions in dissipative systems can have a phase chirp. Thus, the function B in general is complex rather than real. A wide range of exact solutions of the CGLE can be expressed using the ansatz Bx = axexp id logax] ], 14 where the amplitude function ax is real even. This ansatz explicitly uses the parameter d to characterize the phase chirp. We assume that the parameters f j, other than d, are contained in ax implicitly. Now if f is a parameter other than d, then 1 ψ iψ f = i + d a 15 a f f = 4d a β f ax 1 + d2 a x ] 2 ] δax ɛa 3 x μa 5 x dx. 16 The Lagrangian is given by Eq. 3 with the use of 7, where P can now be simplified to P = 21 + d 2 a x ] 2 dx. 17 The parameter d needs to be considered separately. We have 1 ψ iψ = logax] 18 d d = 4 + 4β δa 2 x + ɛa 4 x + μa 6 x logax] dx a x ] d 2 logax] dx. 19 Using Eq. 14 further specifying ax = b sechcx, we find the Lagrangian: L = b2 3b 2 16b 4 ν + 15 c d 2 D 6ω ]. 2 45c The balance equation is now given by Q z = 1 4b 2 5βc d δ + 1b 2 ɛ + 8b 4 μ ]. 15c 21 Derivatives of the Lagrangian with respect to parameters involved can further be calculated. We derive, using Eqs. 4 16, b = d b Q z =, 22 since the energy, Q, does not change for a stationary solution. Hence 5 c d 2 D 6ω 2b 2 16b 4 ν =, 23 then d = 2 3 b2 cdd. 24 However, 1 ψ iψ = d + ix tanhcx 25 c which gives δ c = 2db2 c 2 + ɛ b2 3 + μ8b ] 3 βc2 1 + d Further, we need the derivative of the Lagrangian with respect to the parameter d. It follows, from Eq. 18, 1 ψ iψ d = log b sechcx 27 which, using Eq. 19, gives d = δd 2 + ɛd 4 + μd 6 + βd s ], 28

4 94 A. Ankiewicz et al. / Optical Fiber Technology where the coefficients d i are given by d 2 = 4 a 2 log b sechcx dx = 4b2 1 + log2b, 29 c d 4 = 4 a 4 log b sechcx dx = 4b log2b, 3 9c d 6 = 4 a 6 log b sechcx dx = 8b log2b, c, finally, for d s,wehave d s = 4 a x ] d 2 log b sechcx dx = 4 9 b2 c 1 + 4d d 2 log2b. 32 Now, we are equipped with all necessary relations in order to apply the general formalism to find various particular solutions of the CGLE. The solutions are limited by our ansatz Eqs. 7 14, of course. For the cubic CGLE, it gives the complete set of bright solutions of the CGLE, while, for the cubic quintic CGLE, it allows us to pick up only a small subclass of the solutions. However, thus far, this ansatz provides the only known way to obtain exact solutions for the cubic quintic CGLE. What follows below are a few examples which are particular cases of this general ansatz. 3. Basic NLSE soliton For illustrative purposes we consider the case when all coefficients in 1 are zero except D = 1. The nonlinear Schrödinger equation is a particular case of the CGLE 1 when the dissipative terms the term with ν are zero. This is the simplest case where we can apply the Lagrangian approach. Using the ansatz with ν =, d =, i.e., ψ = b sechcx exp iθz, 33 with arbitrary parameters b c, we find that the Lagrangian is L = b2 2b 2 + c 2 6 θ z. 34 3c So, the condition b = 35 leads to 4b 2 + c 2 6θ z =, 36 while c = 2b2 + c 2 + 6θ z =. 37 Now Q z = is automatically satisfied, so we have only one degree of freedom. We get b = c θ z = ω = c 2 /2. This is the exact one-soliton solution of the NLSE, ψ = c sechcx expic 2 z/2. 4. Cubic CGLE chirp-free soliton Here d =, D = 1 ɛ = 2β, ν = μ =. Using the same ansatz as above, L is the same as above. Now Q z = requires 4b 2 β βc 2 +3δ =, so we no longer have a degree of freedom in the solution: θ z = ω = δ 2β, b= c = δ/β. 38 These are the correct values, as seen from Eq on p. 279 of 14], which was derived in quite a different way. 5. Solution with arbitrary amplitude We know 23] see also Chapter 13 of 14] that when δ =, both cubic cubic quintic CGLE admit soliton solutions with arbitrary amplitude. Thus, setting δ =, ν = μ =, D = 1, using relation of 14] to relate β ɛ, we can directly solve the equations involving Q z the derivatives w.r.t. b c to get 1 + 4β d = 2 1, 39 2β ω = dc β2 4 2β b 2 = dc β 2ɛ 2, 41 with c being arbitrary. Setting b = GF c = G reproduces the known results Section of 14]. Substitution shows that these values also satisfy the d equation. 6. General cubic CGLE soliton Here we only set ν = μ =. We use Eq. 11 for b, c Q z 42 to get 3δD + βd b = β 2Dɛ 3βdɛ, 43 c = 3δɛd d 2 2Dɛ + β3ɛd 1 44 θ δ4βd + D3 + dɛ z = 4Dɛ + 2β3dɛ 1. 45

5 A. Ankiewicz et al. / Optical Fiber Technology On converting, the D = 1 set agrees with the values given in Section of 14]. We then use d = 2b2 cd/3 46 to get the equation for d: 2β Dɛd 2 + 3dD + 2βɛ+ 2Dɛ 2β =. 47 Solving this quadratic equation for D = 1 gives the correct values for d i.e., Eq of 14]. This shows that, to obtain d = chirpless solution, we need Dɛ = 2β. This explains why we cannot just have arbitrary values of ɛ, β for the solution of Eq. 38, which assumes d =. 7. Rational solution Here we set D = 1, δ = seek a rational solution which has the form of Eq. 14 with f 1 ψ 2 = fx= a 2 x = 1 + kf x2 Then 1 ψ iψ = a2 f 1 2f1 2 d + i kf1 2 x2 1, 49 while 1 ψ iψ k = a2 2 d + if 1x ψ iψ d = 1 2 log fx. 51 Hence k = f 1 Re a 2 i + dx 2 δ ψ 2 + ɛ ψ 4 + μ ψ 6 + βψ xx ψ ] dx, 52 similarly for the other 2 parameters. We find / f 1 = πd/4 k ɛ + f 1 μ, while / d involves log terms. Clearly, ψ 2n = fx n, n = 1, 2,... With this solution, we have δ = hence the frequency offset ω is zero. Also, μ is not arbitrary, but is related to ν by a particular expression Eq of 14]. The Lagrangian is L = πf f1 k + d 2 k 2ν, 53 k so that = π f f1 k + d 2 k 2ν, 54 k etc. Solving the equations for k f 1 simultaneously gives 2ν k = 3 + 8βd d 2, f 1 = 3 + 8βd d2 dɛ 1 2ν1 + βd1 + d When we additionally use the equation giving Q z, we get the quadratic equation for d arrive at the correct value Eq of 14]. On conversion, k,f 1 also agree with published values. 8. Quintic CGLE chirp-free soliton One of the soliton solutions of the cubic quintic CGLE is the chirp-free d = soliton. It appears when we take ɛ = 2β μ = 2βν 14]. For convenience, we take β>, ν>, δ<. Writing exp iωz for the z-dependence, we use the ansatz ax = c 1 + g cosh x 2c ] 1/2 56 for the amplitude function. Then we try to find the correct values of c,g,ω for the solution. The result for S 2 is S 2 = Q = 2 2c A, 57 g 1 where A = arctan 1 2/1 + g. We also need the expression for S 4 S 4 = c 3/2 ] 2 g 1 2A 58 g 1 S 6 = 1 c 5/2 ] 3 g ga g 1 The remaining term is P = 1 c 3/2 2 ] g 1 2gA. 6 2 g 1 Then, setting the algebraic part of Q z to zero gives 3g 3 4cν =, 61 so g = cν. Then the transcendental part with function arctan gives βc + 2δ = c = 2 δ >. 62 β Now taking D = 1 using / c = shows that ω = δ 2β. 63 With these values c =, =, 64 g so we have an extremum of L. Thus g = 1 8δν/3β > 1. A similar example, with a different derivation, can be seen from Eqs on p d = of 14]. 9. Lagrangian technique pulsating dissipative solitons Thus far, we have applied a Lagrangian approach to derive exact solutions of various forms of the CGLE. However, not all soliton solution admit an exact analytical form. Some of the solutions of the cubic quintic GLE are known only numerically 24]. These include stationary solitons as well as pulsating ones. The Lagrangian approach can also be used to find the best approximation for these objects when we do not expect to find exact solutions. Approximating stationary solitons with trial functions is a trivial task. This can be done in a similar

6 96 A. Ankiewicz et al. / Optical Fiber Technology fashion to examples of conservative systems 25]. In this section, we will apply the technique to finding approximate CGLE soliton solutions which are not necessarily stationary. It is well known 24] that the CGLE admits pulsating solitons, in addition to solutions in the form of stationary solutions. These correspond to limit cycles of the dissipative dynamical system rather than to fixed points. Thus, these are more complicated objects in the infinite-dimensional phase space. There are no known exact solutions for these formations. However, pulsating solutions can be approximated using various localized z-dependent functions with several parameters the so-called method of moments. Thus, in contrast to the previous sections, we are concerned here with approximate solutions rather than exact ones. In particular, in this section, we show how the Lagrangian method can be used to derive the results of the method of moments 22]. To be specific, we will use a Gaussian-type ansatz Qz ψx,z= A fz exp x2 f 2 z x4 mf 4 z + iczx 2 iθz, 65 where / A 2 = 2 e m/4 m mk 1/4, 4 with K being the modified Bessel function of the second kind. The soliton profile is localized in x has explicit z- dependence. The pulse evolution is described through the variation of total energy Qz, pulse width, fz, chirp factor cz the overall phase, θz. For clarity, we restrict ourselves to the lowest-order Gaussian thus take m =,soa = 2/π 1/4. This leads to the Lagrangian L, where L Q = D 2f 2 + D c2 f 2 Q 2 2 πf 2νQ2 3 3πf f 2 c z θ z. 66 Then, for any function depending on θ in the exponential as in Eq. 65, we have =. 67 θ However, θ = Qz, 68 z so that d dz θ z = Q z. 69 Thus, we reproduce balance equation 4 using the explicit dependence on phase. Consequently, Eq. 69 is the same as the derivative of energy w.r.t. z from the method of moments 22]. Now / c = Dcf 2 Q. This quantity was denoted by I 3 /i in 22]. Further / c z = 1 4 f 2 Q which was denoted by I 2. Also 1 iψ ψ c so that d 1 dz 4 f 2 Q = 2Re = x 2, 7 cf 2 Q x 2 δb 2 + ɛb 4 + μb 6 + βψ xx ψ ] dx, 71 where B = ψ. Here, the left h side equals d/dzi 2 + ii 3 so we have reproduced the second-order moment equation from 22]. Hence f z = 2β fz + 2Dcf 2βc2 zf 3 ɛq 2 π 4μQ πf see Eq. A1 in 22]. Now 1 ψ iψ = i Q 2Q, 73 thus leading to =, 74 Q while 1 ψ iψ = i f 2f 3 f 2 4x 2 75 so / f. Using these relations, we find c z = 2D fz 4 2Dc2 8β c f 2 z Q πf 3 z 8νQ2 3 3πf 4 z see Eq. A13 in 22] 76 θ z = D f 2 z 2βc 5Q 4 πfz 8νQ2 3 3πf 2 z. 77 Equations 72, comprise the three-dimensional dynamical system that governs the evolution of the soliton parameters. This dynamical system is the same as that obtained using the method of moments 22], this is also true when we use m = 1. It has solutions in the form of fixed points limit cycles. The former correspond to stationary solitons, the latter to pulsating solitons. To increase the accuracy of the approximation, we can use a trial function with more parameters derive equations for higher-order moments. If the chirp term is changed to czx n, then 1 ψ iψ c = x n. 78

7 A. Ankiewicz et al. / Optical Fiber Technology Then we have / c cf 2n 2 Q / c z f n Q, hence an nth-order moment-type equation is obtained. So the reduced dynamical system for solitons that was obtained using the method of moments can also be obtained using the Lagrangian approach. When using the same trial function, we obtain the same dynamical system. In this sense, the two techniques are equivalent although the principles used to obtain them are different. 1. Conclusions In conclusion, we have shown that the technique of minimization of the Lagrangian can be used for the dissipative systems described by the complex cubic quintic Ginzburg Lau equation. The technique allows us to derive exact solutions of this equation as well as approximate solutions in the form of pulsating solitons. In the former case, we recover the complete set of known solutions of the CGLE while in the latter case we show that the results obtained using the Lagrangian approach are the same as those obtained using the method of moments. Acknowledgment The authors gratefully acknowledge the support of the Australian Research Council. References 1] N. Akhmediev, A. Ankiewicz, Dissipative solitons in the complex Ginzburg Lau Swift Hohenberg equations, in: N. Akhmediev, A. Ankiewicz Eds., Dissipative Solitons, Springer, Heidelberg, 25. 2] N. Akhmediev, J.M. Soto-Crespo, M. Grapinet, Ph. Grelu, Dissipative soliton interactions inside a fiber laser cavity, Opt. Fiber Technol ] J.N. Kutz, Mode-locking of fiber lasers via nonlinear mode coupling, in: N. Akhmediev, A. Ankiewicz Eds., Dissipative Solitons, Springer, Heidelberg, 25. 4] U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, F. Lederer, Dynamics of dissipative temporal solitons, in: N. Akhmediev, A. Ankiewicz Eds., Dissipative Solitons, Springer, Heidelberg, 25. 5] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations Inverse Scattering, London Mathematical Society Lecture Notes Series, vol. 149, Cambridge Univ. Press, Cambridge, ] N.J. Zabusky, M.D. Kruskal, Interaction of solitons in a collisionless plasma the recurrence of initial states, Phys. Rev. Lett ] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, San Diego, CA, ] L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, ] N. Akhmediev, A. Ankiewicz, R. Grimshaw, Hamiltonian versus energy diagrams in soliton theory, Phys. Rev. E ] N. Akhmediev, A. Ankiewicz Eds., Dissipative Solitons, Springer, Heidelberg, ] I.S. Aronson, L. Kramer, The world of the complex Ginzburg Lau equation, Rev. Mod. Phys ] E.P. Ippen, Principles of passive mode locking, Appl. Phys. B ] Z. Bakonyi, et al., Dissipative solitons their critical slowing down near a supercritical bifurcation, J. Opt. Soc. Am. B ] N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear Pulses Beams, Chapman & Hall, London, ] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw Hill, ] A.D. Bonderson, M. Lisak, D. Anderson, Soliton perturbations: A variational principle for the soliton parameters, Phys. Scripta ] A. Ankiewicz, N. Akhmediev, G.D. Peng, P.L. Chu, Limitations of the variational approach for soliton propagation in nonlinear couplers, Opt. Comm ] G.D. Peng, P.L. Chu, A. Ankiewicz, Soliton propagation in saturable nonlinear fibre couplers, Int. J. Nonlinear Opt. Phys ] A.D. Boardman, L. Velasco, Gyroelectric cubic quintic dissipative solitons, IEEE J. Select. Top. Quantum Electron ] M. Manousakis, et al., Propagation of chirped solitary pulses in optical transmission lines: Perturbed variational approach, Opt. Comm ] S. Chavez Cerda, et al., A variational approach of nonlinear dissipative pulse propagation, Eur. J. Phys. D ] E. Tsoy, A. Ankiewicz, N. Akhmediev, Dynamical models for dissipative localized waves of the complex Ginzburg Lau equation, Phys. Rev. E ] N.N. Akhmediev, V.V. Afanasjev, Novel arbitrary-amplitude soliton solutions of the cubic quintic complex Ginzburg Lau equation, Phys. Rev. Lett ] N. Akhmediev, J.M. Soto-Crespo, G. Town, Pulsating solitons, chaotic solitons, period doubling, pulse coexistence in mode-locked lasers: Complex Ginzburg Lau equation, Phys. Rev. E ] D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A