Modulated amplitude waves in the cubic-quintic Ginzburg–Landau equation

Mathematics and Computers in Simulation 69 (2005) 243–256

Modulated amplitude waves in the cubic-quintic Ginzburg–Landau equation S. Roy Choudhury∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA

Abstract In this paper, we begin to develop a theoretical framework for analyzing the strongly amplitude modulated numerical pulse solutions recently observed in the complex Ginzburg–Landau Equation, which is a canonical model for dissipative, weakly nonlinear systems. As such, the article also reviews background concepts of relevance to coherent structures in general dissipative systems (in regimes where such structures are stable and dominate the dynamics). This framework allows a comprehensive analysis of various bifurcations leading to transitions from one type of coherent structure to another as the system parameters are varied. It will also form a basis for future theoretical analysis of the great diversity of numerically-observed solutions, including even the spatially coherent structures with temporally quasiperiodic or chaotic envelopes observed in recent simulations. © 2004 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Coherent structures; Modulated amplitude waves; Cubic-quintic Ginzburg–Landau equation

1. Introduction Numerous attempts have been made to extend the well-developed concept of soliton interactions in integrable, conservative systems [1,2] to more realistic active or dissipative media which are governed by non-integrable model equations. The reason is that the complicated spatio-temporal dynamics of such coherent structure solutions are governed by simple systems of ordinary differential equations, or ∗

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0378-4754/$30.00 © 2004 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2005.01.003

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low-dimensional dynamical systems, rather than by the original complex nonlinear partial differential equation model. There are situations [2–4] where this approach is appropriate, particularly where the dynamics of various active or dissipative systems is primarily governed by localized coherent structures such as pulses (solitary waves) and kinks (fronts or shocks). Since such structures correspond to spatial modulations, they are also known as spatially localized ‘patterns’. The speeds and locations of the coherent structures may vary in a complex manner as they interact, but their spatial coherence is preserved in such situations. It is tempting to apply this approach to any system which admits pulse and/or kink solutions, but caution is necessary. Coherent structures may be transitory [5,6] when they are unstable to small disturbances in their neighborhood. Also, only some of them may actually be selected due to such stability considerations. This area has become a large and diverse one over the past decade. In this paper, we shall not attempt a comprehensive treatment. Rather, we shall survey only some important salient features of the field by focusing on some recently obtained classes of complicated amplitude-modulated pulse solutions of the complex Ginzburg–Landau equation (abbreviated as the CGLE and also known as the Newell–Whitehead equation in fluid dynamics). This equation is a canonical model for weakly nonlinear, dissipative systems [7–10] and, for this reason, arises in a variety of settings including nonlinear optics, fluid dynamics, chemical physics, mathematical biology, condensed matter physics, and statistical mechanics [7–10]. It also displays many of the generic features of the dynamics of coherent structure patterns in such dissipative models, thus making it a suitable choice for our survey purposes. The remainder of this article is organized as follows. In Section 2, we briefly review some wellknown properties of the coherent structure solutions of this equation. Section 3 summarizes some basic categories of numerically determined pulse solutions, with complicated dynamical behaviors. It also considers a simple dynamical systems approach to recovering these pulse solutions using a coherent structure formulation. In Section 4, we review the sequence of bifurcations in the cubic CGLE model in an attempt to understand and delineate the domain of existence of the complex pulse-type solutions. Section 5 develops the theoretical framework for the analysis of the full cubic-quintic model. Following this, we conclude with some comments about other approaches that have been brought to bear on this problem, as well as additional treatments that may be worth attempting. 2. Background properties of coherent structures of the CGLE equation We shall begin with the canonical governing equation for dissipative systems, i.e. the Ginzburg–Landau equation with both cubic and quintic nonlinear terms or the so-called cubic-quintic CGLE ∂t A = A + (1 + ic1 )∂x2 A + (1 + ic3 )|A|2 A − (1 − ic5 )|A|4 A.

(2.1)

In this section, we shall briefly review some standard background results on the dynamics of coherent structure solutions of this equation closely following the treatment in [3]. These will be necessary in the subsequent sections. The reader is referred to [3] for further details. The primary physical question we wish to address is: suppose we start with the spatially uniform state A = 0 at t = 0, and introduce a small localized perturbation or disturbance. What will the system solution evolve to at long times? The most likely outcomes are: (i) The system decays back to A = 0.

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(ii) A localized pulse, or train of pulses, is formed. If so, is their speed zero, constant, periodic, or chaotic? (iii) A stable, finite amplitude state grows by the creation and propagation of a front which invades the A = 0 state. In this case, we need to determine the wavevector, speed, and frequency of the finite amplitude front. (iv) The system becomes chaotic everywhere. Note that we shall concentrate on the coherent structure states and not focus on the chaotic or turbulent states admitted by (2.1). One physical reason is that we shall attempt to understand very interesting classes of numerically observed coherent structure solutions. Another is that it is the coherent structure solutions which are of potential interest as information carriers in areas such as nonlinear optical transmission (in a manner analogous to solitons for integrable systems). We shall first consider uniformly translating coherent structure solutions of the form ˆ − vt) A(x, t) = e−iωt A(x (2.2a) ˆ A(ξ) = a(ξ)eiφ(ξ)

(2.2b)

ξ ≡ x − vt

(2.2c)

In particular, note that the structures in (2.2) may have their amplitudes varying in an arbitrary manner with respect to the traveling-wave variable, or pseudo-time, ξ. Also, the traveling wave speed v could be generalized to be a function of ξ for more general coherent structure solutions. The quantities v and ω are eigenvalues and are determined by the initial conditions for large or ‘spatially extended’ systems, as well as boundary conditions for spatially moderately extended systems. We shall define pulses or solitary waves as structures which have the same values at both ends of the ξ axis, and fronts as ones which have distinct values at the two ends. In particular, further differentiations are sometimes made between simple fronts (connecting a zero value at one end of the ξ axis to a nonzero value at the other) and domain walls or nonlinear fronts (connecting different nonzero value at the two ends). We shall not consider this in any detail in this introductory treatment. Nor shall we discuss the distinction between sources and sinks based on the signs of the group velocity at the two ends. Defining the variables q(ξ) ≡ ∂ξ θ

(2.3a)

κ(ξ) = ∂ξ a/a

(2.3b)

insertion of (2.2) in (2.1) yields the ordinary differential equations ∂ξ a = κa

(2.4a)

∂ξ q = G(a, q, κ)

(2.4b)

∂ξ κ = H(a, q, κ)

(2.4c)

G = −b˜ 1 (ω + vq) + c˜ ( + vκ) − 2κq − (b˜ 1 c3 + c˜ 1 b3 )a2 − (b˜ 1 c5 + c˜ 1 b5 )a4 H = −˜c1 (ω + vq) − b˜ 1 ( + vκ) + q2 − κ2 + (b˜ 1 b3 − c˜ 1 c3 )a2 + (b˜ 1 b5 − c˜ 1 c5 )a4

(2.5a)

with

(2.5b)

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b˜ 1 = b1 (b12 + c12 )−1

(2.5c)

c12 )−1 .

(2.5d)

c˜ 1 =

c1 (b12

+

Following the terminology in [3], we may classify the fixed points of (2.4) as: (i) the linear fixed points L aL = 0,

κL = 0

(2.6)

and (ii) the nonlinear fixed point(s) N aN = 0,

κN = 0.

(2.7)

In general, there are two linear fixed points, while the number of nonlinear fixed points may vary from zero to four [3] depending on the parameters. Reference [3] considers the stability of these fixed points, the multiplicity of the homoclinic and heteroclinic orbits joining them, limited analytic solutions, perturbative approaches, and numerical solutions to consider the coherent structures of (2.1). An additional idea developed in [3] is the concept of ‘linear marginal stability’ and its use in formulating selection criteria for the final selected coherent structure state to which localized initial conditions evolve. The reader is referred to the original paper for further details. Having laid out some necessary background for the coherent structure solutions, we shall now begin considering some intriguing pulse coherent structures of (2.1) recently obtained via numerical simulations.

3. Numerical pulse solutions Very interesting classes of pulse solutions of (2.1) have recently been obtained numerically [12]. Note that there are additional numerical solutions, such as spatially chaotic or turbulent states, which are not considered here. As stated earlier, our primary focus is on attempting to analyze the stable coherent structure solutions which may be considered as the basic building blocks for the dynamics of the system. They may also be regarded as possible information carriers at some future date in the same way as solitons in the context of integrable systems. We shall primarily discuss the solutions found in the context of Nonlinear Optics [13,14]. However, they have also been observed in other numerical simulations, as well as in experiments. In particular, Akhmediev et al [12] consider the cubic-quintic CGLE in the optical form (with space and time switched): D (3.1) ψtt + |ψ|2 ψ + ν|ψ|4 ψ = iδψ + i|ψ|2 ψ + iβψtt + iµ|ψ|4 ψ. 2 Note that the parameters in this equation are different from those in (2.1), but the connections between the two are clear from a direct comparison. Note too that two parameters in (2.1) have been scaled to unity, as may always be done (see [3] for details). Akhmediev et al. observe plain pulses or solitary waves with constant amplitudes. However, and more interestingly, they observe spatially localized coherent structures with pulsating amplitudes (‘pulsating solitons’, Fig. 1), and some of these double and then quadruple in period as the parameter  is varied iψz +

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Fig. 1. Pulsating solitary wave of (1) for D = 1,  = 0.66, δ = −0.1, β = 0.08, µ = −0.1, and ν = −0.1.

(Fig. 2). In addition, they observe interesting multi-peaked coherent structure solutions, such as the ‘creeping soliton’ (Fig. 3) and the ‘slug’ (Fig. 4). Note that the amplitudes vary in a complex manner in the last two figures, but the structure is still spatially localized in a coherent fashion. In addition, Akhmediev et al have carried out a systematic analysis of the parameter space to classify the domains where various kinds of coherent structures are observed. Two typical examples are shown in Figs. 5 and 6 in the (ν, ) and (β, ) planes respectively. Note the dark areas in each figure corresponding to the period doubling and period quadrupling referred to earlier. Although not stated, in these regions it is clear that one has Feigenbaum’s infinite sequence of period doublings eventually leading to chaotically varying envelopes. More will be said about this subsequently in this and the following section. One may attempt to directly analyze such structures within the coherent structure formalism outlined in the previous section. Note that this formalism is valid since the numerics reveal that we are indeed dealing with long-lived, stable coherent entities which are principal organizing centers for the dynamics. Also note that, in order to treat the case of the ‘slug’ in Fig. 4, one would need to generalize ansatz (2.2) and the resulting low-dimensional system of ODEs (2.4) to allow the translation speed v of the coherent structure to vary with ξ. The details of this analysis are too long and involved for this article and will be published elsewhere. However, we may illustrate the idea concretely and simply by considering the following ODE (obtained by a traveling-wave or coherent structure reduction of a PDE using notation analogous to (2.2)): dA = −µA + iA + iFA∗ + iβ|A|2 A + N|A|2 A + iF!(A3 + 3|A|2 A∗ ) − ih|A|4 A. dξ

(3.2)

This equation may be split into real and imaginary parts and numerically solved. For each set of parameters, the numerical solutions for the amplitude A may then be categorized as limit cycles (periodic), period doubled limit cycles, quasiperiodic, or chaotic using standard numerical diagnostics [14,15]. A typical plot summarizing the nature of the solutions is shown in Fig. 7 in the (, F ) plane and for specific values of the other parameters. Note the regions of regular or plain pulses (stable fixed points), periodically pulsating solutions (stable limit cycle behavior of the amplitude or envelope A, corresponding to periodically

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Fig. 2. Period doubling of pulsating solitary waves of (1) with D = 1, δ = −0.1, β = 0.08, µ = −0.1, and ν = −0.07 as  is varied. The  values are: (a)  = 0.75, (b)  = 0.785, and (c)  = 0.793.

Fig. 3. Creeping solitary wave of (1) for D = 1,  = 1.3, δ = −0.1, β = 0.101, µ = −0.3, and ν = −0.101.

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Fig. 4. ‘Slug’ solitary wave of (1) for D = 1, δ = −0.1, β = 0.08, µ = −0.11, ν = −0.08, and  = 0.835.

pulsating solitons of the PDE), and period doubled limit cycles (corresponding to period-doubled pulsating solutions). Note the similarity of many features with those in Figs. 5 and 6. Clearly, an analogous analysis of the numerical solutions of Akhmediev et al summarized in Figs. 5 and 6 may be carried out in a similar manner using the coherent structure formulation in Section 2 (and allowing the traveling wave speed v to vary with ξ). Having considered numerically obtained families of complex pulse coherent structure solutions of (2.1), let us next turn to a variety of other approaches to this problem.

Fig. 5. Plain pulses, pulsating solitary waves (enclosed in heavy lines), period doubled solutions (gray area), and chaotic solutions in the (ν, ) plane for D = 1, δ = −0.1, β = 0.08, and µ = −0.1.

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Fig. 6. The same as Fig. 5 in the (β, ) plane for D = 1, δ = −0.1, µ = −0.1, and ν = −0.08.

Fig. 7. Regions of plain pulses (A), periodically pulsating solitary waves (B), and period doubled pulsating solitary waves (C) of (13) in the (, F ) plane for µ = 1, β = 0.4179, N = 0.1739, h = −0.0973, and ! = 0.0278.

4. Bifurcations in the cubic CGLE and various theoretical approaches 4.1. Bifurcations in the cubic CGLE In this section, we shall briefly summarize some of the important bifurcations relevant to the numerical solutions of the previous section. We shall also briefly consider some theoretical approaches which have been brought to bear on this problem. In keeping with the intended level of the article, we shall again omit many of the more technical details. The interested reader is advised to consult the primary references, as well as the general comprehensive technical review of the cubic CGLE in [8] for further details and additional solution behaviors. Once again, our primary focus here is on the pulse coherent structures.

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Considering a simple stationary pulsating solution A(x, t) = eic3 t

(4.1)

one may show that this undergoes a so-called modulational, or Benjamin–Feir, instability for c1 c3 ≥ 1.

(4.2)

A more relevant instability is however that of plane wave solutions. The simplest relevant solutions of the coherent structure ODEs (2.3) correspond to (a, q, r) = ((1 − r2 )1/2 , r, 0) corresponding to plane-wave solutions (of wavenumber r) A=




1 − r 2 ei(rx−ωr t)

(4.3)

with angular frequency ωr = −c3 + q2 (c1 + c3 ).

(4.4)

A linear stability analysis of these solutions may be performed by considering the perturbed solution


˜ r = ( 1 − r 2 + δa)ei(rx−ωr t) A σt ikx

δa α e e .

(4.5a) (4.5b)

Using these and linearizing in the small perturbation δa, it is straightforward to show that σ(k) = −k2 − 2irc1 k − (1 − r 2 ) ± {(1 + c32 )(1 − r 2 )2 − [c1 k2 − 2irk − c3 (1 − r 2 )]2 }1/2 .

(4.6)

For long-wavelength perturbations (or wavenumber k− > 0), the ‘growth rate’ σ becomes positive (and the perturbation δa begins to grow) for r = rE =



(1 − c1 c3 )/[2(1 + c32 ) + 1 − c1 c3 ].

(4.7)

Note that, unlike the Benjamin–Feir instability where all wavelengths go unstable, only a certain band of long wavelengths are unstable here. This is the so-called Eckhaus instability, and it immediately leads to plane wave solutions with time-dependent amplitudes. Clearly, this instability is relevant to the pulses with modulated amplitudes which were considered in the previous section. In fact, it is straightforward to show that the Eckhaus instability leads to the onset of quasiperiodic solutions. Say that σ = 0 and the perturbations are neutrally stable on the arbitrarily shaped neutral curve [7,10] shown in Fig. 8. As the parameter (say c3 ) is changed and one crosses the neutral curve into the unstable regime, two wavenumbers k1 and k2 simultaneously go unstable. In general, their ratio is incommensurate or irrational, i.e. k1 m = (4.8) k n where m and n are integers. Thus, via (4.5), the perturbed unstable solution contains two modes with wavenumbers of incommensurate spatial periods (wavelengths) and is thus a two-mode quasiperiodic solution. Subsequent secondary bifurcations may occur and may be similarly analyzed. Provided the growth rate σ of the perturbations remains small, one may perform a perturbative weakly nonlinear analysis [7,10] of the Eckhaus instability. Such an analysis [17] reveals that weakly nonlinear effects may sometimes stabilize or saturate the emerging quasiperiodic solutions. These solutions thus

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Fig. 8. A typical neutral curve in the (wavenumber, parameter) space. Note that the region above the curve is assumed to be unstable here, although one may reverse this and the curvature.

co-exist with the original linearly unstable (but nonlinearly stabilized) plane wave (4.3). For obvious reasons, such quasi-periodic stable solutions are referred to as modulated amplitude waves (MAWs). As already noted, the Eckhaus instability thus leads into a regime of modulated amplitude plane-wave solutions that is reminiscent of the modulated amplitude pulse coherent structures which we discussed in Section 3. However, in order to treat general coherent structures with arbitrarily varying amplitudes well into the strongly nonlinear regime of this instability, we must abandon the starting plane-wave solutions in (4.3) and return to the general coherent structure Eqs. (2.2)–(2.5). One may now start from initial conditions corresponding to the initial plane wave, i.e. (a, q, κ) = ((1 − r 2 )1/2 , r, 0) and consider the evolution of the solutions as governed by (2.4)/(2.5). This has been done in [18] using the branchfollowing bifurcation software AUTO. The details are quite technical and are thus omitted here. However, we shall summarize the results since they will give us an alternative perspective on the pulse solutions of Section 3, as well as clearly delineate their region of existence in parameter space. Background material on the various bifurcations we shall encounter may be found in [11,16]. Starting from initial conditions corresponding to the plane wave, i.e., (a, q, κ) = ((1 − r2 )1/2 , r, 0) and varying the parameter c3 , one first has the Eckhaus instability (a Hopf bifurcation) where the smallest wavenumber k = 2π/P, with P the wavelength or spatial period, is destabilized. Two branches now occur (see [18]) and the software adjusts the eigenvalues ω and v automatically to follow them. As discussed above, this Eckaus/Hopf bifurcation leads to the onset of MAWs or localized pulses with modulated envelope amplitudes, i.e. the ‘pulsating’ solitons in the numerical results of Section 3. These MAWs emerge at this bifurcation with zero traveling wave speed v if the value of the average phase gradient parameter (over a spatial period or wavlength P) ν≡

1 P

 0

P

dxφx

(4.9)

is zero, and with non-zero v values otherwise. In the former case, they undergo a subsequent drift-pitchfork bifurcation leading to non-zero v values. Following this, as c3 is increased further they undergo repeated spatial period-doubling bifurcations (as seen in the numerical results in Fig. 2). At each bifurcation, the dynamics is driven to the shorter

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period MAWs since the emergent double-period MAWs have linearly unstable eigenvalues. Also, the MAWs begin to acquire a multi-peaked or multi-humped structure due to these period-doublings. The ‘creeping soliton’ of Fig. 3 is such a multipeaked MAW with v > 0. Similarly, the ‘slug’ in Fig. 4 is one with v being periodic which, as mentioned earlier, may be captured by generalizing the coherent structure formulation of (2.2)–(2.5) to allow variable v. For large or ‘spatially extended’ systems, the system may undergo an infinite sequence of spatial period doublings (as seen in the purely numerical simulations in Figs. 5 and 6) leading to a spatially disordered or random ensemble of MAWs. This is similar to the chaotic regime near the top right corner of Fig. 5 and the bottom left corner of Fig. 6, except that the actual chaotic regime depends on which of the several parameters are being varied, and which are kept fixed. As c3 is increased further, the upper and lower branches meet and terminate in a saddle-node bifurcation (where two fixed points/branches annihilate). This marks the end of the regime of existence of MAWs, or the amplitude-modulated pulses of Section 3. If other parameters are varies, as done in the numerical simulations of Section 3, the actual details of the bifurcation sequence vary somewhat. However, the general picture outlined above remains valid and serves to delineate the regime of existence of the MAWs, or spatially localized pulses with complex amplitude modulations, of Section 3. 4.2. Various theoretical approaches Various other theoretical approaches have been brought to bear upon this problem. We shall mention them only very briefly, since they are quite specialized. The interested reader may refer to [8] for a general review. One may examine the stability of the MAWs using Floquet/Bloch theory [19]. Some series expansions of MAWs have been derived [20], but they do not generalize easily. Linear stability analysis MAWs near the Eckhaus threshold [21], and singular perturbation analyses of their interactions [22] have been performed. There are also normal form analyses of the various bifurcations we have considered above, as well as some spectral analysis describing how the spectrum of the linearized equation around a coherent structure controls its stability, as well as the overall dynamics. Many of these approaches are still being applied to coherent structures in various important dissipative and active pattern-forming nonlinear PDEs.

5. Theoretical framework for the cubic-quintic CGLE Returning now to the coherent structure ODEs (2.4) and (2.5), note that the nonlinear fixed points (2.7) correspond, via (2.2), to traveling wave solutions A(x, t) = aN eiqN t e−iωN t

(5.1)

of the PDE (1) with ωN ≡ ω + vqN

(5.2)

and qN being the value of q at the nonlinear fixed point. These traveling waves will be our basic plane wave solutions of (2.1).

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Using (2.4), (2.5), and (2.7), the nonlinear fixed points of (2.4) satisfy the equations 4 2 (b5 c˜ 1 + b˜ 1 c5 ) + aN (b3 c˜ 1 + b˜ 1 c3 ) − c˜ 1  + b˜ 1 (ω + vq) = 0 aN

(5.3)

4 2 ˜ aN (b5 b˜ 1 − c˜ 1 c5 ) + aN (b1 b3 − c˜ 1 c3 ) − b˜ 1  − c˜ 1 (ω + qv) = 0.

(5.4)

and

Consistency of these equations (identical roots) requires b1 b3 − c1 c3 b3 c1 + b1 c3 = b5 c1 + b1 c5 b1 b5 − c1 c5

(5.5)

2 (b12 + c12 ) −b1  − c1 ωN + qN b1 ωN − c1  . = b5 c1 + b1 c5 (b1 b5 − c1 c5 )

(5.6)

and

Solving the last two equations yields c5 = b5 c3 /b3 and

or

(5.7)

√ b3 v − ! qN ≡ q1 = 2(b3 c1 + b1 c3 )

(5.8a)

√ b3 v + ! qN ≡ q2 = 2(b3 c1 + b1 c3 )

(5.8b)

where ! ≡ −4(b3 c1 + b1 c3 )(−c3  − b3 ω) + b32 v2 .

(5.9)

For qN given by (5.8a)/(5.8b), and κN = 0 (from (2.7)), it is straightforward to solve (5.3) yielding the 2 2 possible values for aN = aNi at the fixed points: (i) For qN ≡ q1 in (5.8a), 2 = ([θ1 − {θ2 (θ3 − θ4 )}1/2 ]/θ5 )1/2 aN1

(5.10a)

2 = ([θ1 + {θ2 (θ3 − θ4 )}1/2 ]/θ5 )1/2 . aN2

(5.10b)

or

(ii) For qN ≡ q2 in (5.8b), with 2 = ([θ1 − {θ2 (θ3 + θ)4}1/2 ]/θ5 )1/2 aN3

(5.11a)

2 = ([θ1 + {θ2 (θ3 + θ4 )}1/2 ]/θ5 )1/2 . aN4

(5.11b)

or

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θ1 ≡ −b3 (2b32 c12 + 4b1 b3 c1 c3 + 2b12 c32 ), θ2 ≡ b32 (2b32 c12 + 4b1 b3 c1 c3 + 2b12 c32 )2 − 4b3 b5 (2b32 c12 + 4b1 b3 c1 c3 + 2b12 c32 ), θ3 ≡ 2b3 c12  − 2b1 c1 c3  + 2b1 b3 c1 ω + 2b12 c3 ω + b1 b3 v2 , 

θ4 ≡ b1 v 4(b3 c1 + b1 c3 )(c3  + b3 ω) + b32 v2 , and θ5 ≡ 2b5 (2b32 c12 + 4b1 b3 c1 c3 + 2b12 c32 ).

(5.12)

As mentioned earlier, each of the four fixed points given by (5.10)/(5.11) corresponds to a traveling wave, or plain pulse solution, of (2.1). The next natural step is to consider when such plain pulse solutions of (2.1) are stable, and hence accessible from various initial conditions within the basin of attraction of the corresponding fixed point. Following standard methods of phase plane analysis, the characteristic polynomial of the Jacobian matrix at each fixed point has the form λ3 + α1 λ2 + α2 λ + α3 = 0

(5.13)

where α1 , α2 , and α3 are complicated expressions dependent on the particular fixed point in question. Using the Routh-Hurwitz criterion [2], the necessary and sufficient conditions for (5.13) to have Re(λ1,2,3 ) < 0, and hence a stable fixed point (or stable plane wave solution of (2.1)), are: α1 > 0,

α3 > 0,

and

α1 α2 − α3 > 0.

(5.14)

When these conditions are satisfied at any of the fixed points given by (5.10)/(5.11), the corresponding plane wave solution of (2.1) given by (5.1) is stable and accessible from initial conditions within the basin of attraction of the corresponding fixed point. Next, we may consider various local bifurcations in the system of Eqs. (2.4). In particular, we shall focus on the occurrence of Hopf bifurcations, rather than saddle-node, transcritical, or pitchfork bifurcations, since the former would give rise to periodic modulations of the amplitude a(ζ). From (2.2) (or (5.1)), we see that these would correspond to plane wave solutions of (2.1) with periodically modulated amplitudes (and phases). Following standard techniques [23,24], the Hopf bifurcation, as evinced by a pair of imaginary eigenvalues in (5.13), occurs at α1 α2 − α3 = 0.

(5.15)

The details of carefully invoking the Hopf bifurcation theorem to show that periodic solutions indeed occur when (5.15) is satisfied will be reported elsewhere. In the context of (2.1), (5.15) represents the transition boundary between plane wave solutions and pulsating plane waves. This may be taken as somewhat analogous to the transition between plain pulses and pulsating solitary waves in the numerical solutions shown in Figs. 5 and 6, although pulse solutions of (2.1) correspond to homoclinic orbits at any of the fixed points of (2.4) (and not to the fixed points themselves, as the plane waves do). Careful numerical investigations of the transition boundary (5.15), as well as analytical treatments of the secondary pitchfork and period doubling bifurcations numerically delineated in Section 4.1 will be undertaken subsequently.

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6. Summary In this article, we have summarized some topical and important numerical and theoretical results on complex pulse solutions of the cubic and cubic-quintic complex Ginzburg–Landau equation. As mentioned earlier, this is a canonical equation for weakly nonlinear, dissipative systems. For this reason, the results have relevance to general dissipative systems, particularly in regimes where their dynamics is dominated by stable coherent structures. Acknowledgement The author would like to sincerely thank Nail Akhmediev for generously sharing his understanding and his numerical results. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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