The cubic–quintic complex Ginzburg–Landau is the amplitude equation for systems in the vicinity of an oscillatory sub-critical bifurcation (Andronov–Hopf), and it shows different localized structures. For pulse-type localized structures, we review an approximation scheme that enables us to compute some properties of the structures, like their existence range. From that scheme, we obtain conditions for the existence of pulses in the upper limit of a control parameter. When we study the width of pulses in that limit, the analytical expression shows that it is related to the transition between pulses and fronts. This fact is consistent with numerical simulations.
In this work, we are interested in systems out of thermodynamic equilibrium, which are extended in space. This is the case for many topics in modern science, such as hydrodynamics, chemical reactions, population dynamics, nonlinear optics and granular media. One experimental example, closely related to our work, is a binary-fluid layer that is heated from below and presents localized pulses (Kolodner et al. 1988; Kolodner 1991).
More specifically, we are interested in bifurcations. Given a system with particular symmetries, the bifurcation sets the partial differential equations that must be used as a model. In our case, we are concerned with an oscillatory sub-critical bifurcation (Andronov–Hopf) that is modelled by the cubic–quintic complex Ginzburg–Landau (CGL) equation (van Saarloos & Hohenberg 1990, 1992; Descalzi et al. 2001). In this model equation, we will be interested in bifurcations between its different stable solutions.
The cubic–quintic CGL equation has demonstrated an extraordinary richness of stable solutions. Many different stable localized structures such as pulses (Thual & Fauve 1988; Deissler & Brand 1994; Afanasjev et al. 1996) and holes (Sakaguchi 1991; Descalzi & Brand 2005; Descalzi et al. 2005) have been observed, as well as coexistence between those structures (Hayase et al. 2004; Descalzi et al. 2006a) and their interactions (Descalzi et al. 2006b, 2007). However, this paper will focus only on the particular case of pulses.
From the pioneering results of Thual & Fauve (1988), many theoretical and numerical studies of pulses have been carried out (Fauve & Thual 1990; Malomed & Nepomnyashchy 1990; Hakim & Pomeau 1991). We can remark on the contribution of van Saarloos & Hohenberg (1990): they established that there is a well-defined range of parameters where pulses can be found, and also, out of that range, the system shows fronts.
Years later, Descalzi et al. (2002, 2003) pointed out that the emergence of pulses takes place via a saddle-node bifurcation at the lower value of a control parameter (Descalzi 2003). This important result comes from an approximation method for pulses (called a matching approach), which is the basis for this work too.
Recently, mathematical conditions for the transition between pulses and fronts (at the upper value of the control parameter) were obtained in order to understand the second limit of the existence range of pulses (Gutiérrez & Descalzi 2007). But at that time, the meaning of those conditions remained unclear. Our purpose now is to give an interpretation of those mathematical conditions and to make comparisons among them and numerical simulations.
2. Approximating stable localized pulses
Our starting point is the cubic–quintic CGL equation, 2.1 where the subscripts x and t denote partial derivatives with respect to space and time, respectively. is a complex field. The control parameter μ is considered as real without losing generality. The parameters β=βr+iβi, γ=γr+iγi and D=Dr+iDi are in general complex and contain the information of the physical problem under study. In order to ensure that the bifurcation is sub-critical and saturates to quintic order, it is necessary to restrict ourselves to βr>0 and γr<0. For the homogeneous case, the system shows coexistence of stable solutions in the range .
If the localized structure is stationary, which means r is a function depending only on the space, then it is possible to make the following ansatz (Thual & Fauve 1988), 2.2 where Ω is an unknown parameter. Replacing the ansatz in equation (2.1) one obtains two real equations, and, after simple algebra, the results 2.3 2.4 with and . It is important to note that, if θ0x is constant, one can explicitly integrate equation (2.3). If the associated numerical simulations are observed (Descalzi & Brand 2005; Descalzi et al. 2005), the validity of the condition on θ0x becomes clear in almost the whole space, except around the centre of the localized structure. This numerical fact allowed Descalzi et al. (2002, 2003) to propose the following approximation scheme (or matching approach): the space is divided in one region near the centre of the pulse (core) and in another region out of the centre of the pulse (outside the core) (figure 1).
Outside the core, where θ0x is constant (+p for x<xc and −p for x>xc), one can explicitly integrate equation (2.3). The solution is 2.5 where 2.6 and a=−3β+/2γ+, b=−3(−μ++p2)/γ+. x0 is a constant that emerges from the translation symmetry of the cubic–quintic CGL equation.
The frequency Ω can be asymptotically evaluated: , which depends only on p (and on the parameters of equation (2.1)).
Inside the core, one can approximate the functions R0(x) and θ0x(x) by their first terms in Taylor expansion writing 2.7 where (Rm,ϵ,α) are unknown parameters. Rm is the maximum height of the pulse at x=0.
The next step in the matching approach is to impose continuity on the functions at x c=−p/α. The continuity in R0(x) fixes the value of x0 in terms of Rm and p, 2.8 where 2.9 and .
The continuity of R0x(x) at x=xc gives us a relation between Rm and p, 2.10
A second relation emerges from a condition of consistency (Thual & Fauve 1988), obtained by multiplying equation (2.4) by R0(x) and integrating in the real axis. Since R0(x) is a symmetric function, the relation is reduced to 2.11
The integrals can be evaluated as in Descalzi et al. (2002). The existence of pulses is restricted by the conditions f(Rm,p)=0 and g(Rm,p)=0 in the following way: pulses exist if the matching between the two regions (core and outside the core) takes place. This occurs when the conditions are simultaneously fulfilled, and graphically it implies the intersection between the curves defined in the (Rm,p) plane (figure 2).
Through this procedure, the mechanism of emergence of pulses has been explained: there is a critical value μc1 at which, for μ < μc1, the curves f=0 and g=0 do not intersect, implying that there are no pulses. For μ>μc1, the curves intersect at two points that represent stable and unstable pulses. This means that the emergence mechanism of pulses is a saddle-node bifurcation (Descalzi et al. 2002, 2003).
3. Transition from pulses to fronts
(a) Mathematical conditions
As demonstrated in Gutiérrez & Descalzi (2007), from a mathematical point of view, the disappearance of pulses is related to R0, which is assumed to be real. If μ is increased above the critical value μc2, R0 is no longer real. For that reason, pulses cannot exist. The region in the (Rm,p) plane, where R0 does not exist as a real variable, will be called R (figure 2). When μ is increased, R invades the (Rm,p) space. If the intersection of the curves f=0 and g=0 (in blue and purple, respectively) takes place in R, the pulse does not exist. That is the case of figure 2c (zoom in figure 2f).
In conclusion, there is a qualitative change in the system at μc2, where pulses stop existing. At μc2, the intersection of f=0 and g=0 takes place at the border of R, which is a straight line defined by the condition u2*=0 (it implies a2=4b, or ), which fixes p in the constant value pc 3.1 with
In the dispersion-less case of Di=0 and Dr=1, coefficients α are reduced to (μ of Maxwell), α2=α3=0, and expression (3.1) takes the simpler form 3.2
To obtain μc2, we explicitly write f(Rm,p,μ) and g(Rm,p,μ), and then remove p taking into consideration that pc depends only on μ, as can be seen in equation (3.1). In this case, when p tends to pc, the expression of g takes the form (more details are given in Gutiérrez & Descalzi (2007)) 3.3
The above expression depends only on μ. Then μc2 can be obtained explicitly, 3.4 where
If we take the dispersion-less limit Di=0, using Dr=1 and defining , we get a simpler expression of μc2, which is 3.5
(b) Analytical prediction and numerical simulations
Equation (2.1) has seven parameters, but only four of them are relevant, because we can re-scale amplitude A, space x and time t. As a particular case, we can fix other parameters such as γi=Di=0 and only two are necessary to describe the system. We will use μ as before, because it is the control parameter. As the second parameter, we will use βi because it is the only non-variational parameter (Dr=1 for simplicity).
The phase diagram (μ,βi) is not simple a priori. From a numerical study, it is possible to observe a lot of different localized solutions, as can be seen in Descalzi et al. (2005). In this case, our goal is simpler. It is to compare numerical simulations with the limit for the existence of pulses given by equation (3.5), which is obtained from the matching approach.
Rewriting the earlier mentioned equation, we obtain 3.6 and can plot a curve in the space (μ,βi), which can be seen as the blue solid line on the right in figure 3.
For numerical computations, we use RK4 with dt=0.1, dx=0.4 and 600 points to get a spatial box of L=240. Localized structures are selected by the system, given the parameters and an initial condition. We use the initial condition in phase (Descalzi et al. 2005). In figure 3, the green dots correspond to numerically simulated pulses. At the transition to fronts, their existence boundary is in good agreement with the curve obtained from equation (3.6) (blue solid curve).
(c) Width of pulses
As can be seen from figure 4, if μ is increased near the critical value μc2, the width of pulses enlarges abruptly. Then, studying the width of pulses could be useful to characterize the transition from pulses to fronts, and it can be done using the matching approach. One possible way to define the width Δ is by measuring it at the half height of the pulse, 3.7 where xm is the half width, defined to satisfy R0(xm)=Rm/2. Then it is equal to 3.8 with
If we use expression (2.8) of x 0, we obtain another equation for the half width 3.9 which contains the same singularities of x 0 (since sm does not reach zero in the whole range explored by us). In particular, when u* tends to zero (at μc2), xm diverges.
Near μc2, the singularity of xm shows that, in the transition to fronts, pulses become wider. This is presented in figure 5 (grey dashed line) and is compared with the numerical simulation (solid line). The width of pulses going to infinity is suggested in both computations. In the numerical case, infinite width cannot be observed since the numerical method (RK4) has finite resolution. Using dt=0.1 (our case), the parameters have a resolution of 10−4.
In order to get bifurcation diagrams (figure 6), we add unstable pulses to the picture of stable pulses. Both exist since the bifurcation in the cubic–quintic CGL equation is sub-critical.
Considering both kinds of pulses, we re-obtain that their emergence takes place via saddle-node bifurcation (figure 6b). With regard to the disappearance of pulses, it can be said that the stable ones lead to fronts at μc2 (increasing their width to infinity), and the width of unstable pulses also tends to infinity at μ=0, but its amplitude tends to zero, collapsing with the homogeneous state A=0 (figure 6c and d).
Note that we are assuming that the stationary pulse obtained with the matching approach is stable for all μ∈[μc1,μc2]. Our numerical study, keeping only the non-variational parameter βi, is in good agreement with this assumption. However, it has been reported in a high dispersion case (Tsoy et al. 2006), where pulses have a region of pulsating behaviours. Eventually, the stationary pulse undergoes a Andronov–Hopf bifurcation before the transition to front. This type of instability could give rise to more complex bifurcation diagrams (Chang et al. 2007), including chaotic behaviours.
(d) Geometrical picture
The bifurcation scenario that we are proposing for the transition from stationary pulses to fronts can be depicted in a more geometrical fashion. Actually, we can rewrite equations (2.3) and (2.4) as a three-dimensional dynamical system 3.10 3.11 3.12 where S=θ0x. Then, the stationary pulses that we analysed using the matching approach corresponds to a heteroclinic orbit of the earlier dynamical system, which links the hyperbolic points (R0,W,S)=(0,0,p) and (R0,W,S)=(0,0,−p). The value of p is obtained by the matching procedure.
Therefore, the stationary pulses are like a ‘particle’ that follows a trajectory (the heteroclinic orbit) in the phase space (R0,W,S) of the dynamical system (3.10), (3.11) and (3.12), as the ‘time’ x runs. As μ increases and approaches the critical value μc2, the particle spends more time near the point (R0(xc),R0x(xc),p), after it leaves the hyperbolic points (0,0,p) (i.e. outside the core). Then, it soundly changes from p to −p (inside the core) and repeats the same pattern for −p, until it converges to the hyperbolic point (0,0,−p). In the limit , the time spent near the points (R0(xc),R0x(xc),±p) tends to infinity.
At the point μ=μc2, the dynamical system (3.10), (3.11) and (3.12) has a heteroclinic orbit that links the hyperbolic points (0,0,pc) and (as well as for −pc), where 3.13 for that reason , which corresponds to the Maxwell point of equation (2.3) for a fixed S≡pc. However, this orbit cannot be obtained from a pulse generated for μ<μc2. Moreover, it cannot be obtained numerically, because it demands an infinite size system. If the system has a finite size, we always need a source of waves, which is in fact the core of the pulse.
What we observe numerically is that the width of the pulse tends to infinity when μ increases and approaches the critical value μc2. For μ>μc2, we only observe a propagative behaviour, which is not achievable by equations (2.3) and (2.4) or equations (3.10)–(3.12), which are restricted for static solutions.
We were concerned with systems that are modelled by the cubic–quintic CGL equation. This equation accepts different stable solutions such as fronts and localized structures (pulses and holes). We studied analytically the transition from pulses to fronts.
More specifically, we reviewed an approximation scheme for pulses that give us mathematical conditions for their existence in the parameter space. Using that scheme, we obtained an analytical expression associated with the disappearance of pulses. In a particular case, we checked the analytical expression with numerical simulations by doing a phase diagram, and they are in good agreement.
In order to see what happens beyond pulses, we explored their width and saw that it diverges at the expected critical value. The divergence in the width was observed in numerical simulations too. Then, we concluded that pulses disappear giving rise to fronts. Because the width is a good variable to describe pulses, we constructed bifurcation diagrams. One of them, related to pulse emergence, is consistent with a previously known result.
It is important to emphasize that the scenario proposed in this paper is related to the stability of the stationary pulse, which is what is observed in most of the dispersion-less cases. In high-dispersion cases, a more complicated scenario is expected (Chang et al. 2007).
Finally, we conclude that with a simple approximation scheme for pulses, the matching approach, we can get important features of their dynamics.
The authors wish to acknowledge the support of FAI (Project ICIV-003-08, Universidad de los Andes, 2008), FONDECYT (Project No. 1070098), FONDECYT (Project No. 3070013) and Project Anillo en Ciencia y Tecnologia ACT15. The numerical simulations were done using the software DIMX, developed in the Institut Non Linéaire de Nice, France. The authors also thank Jaime Cisternas for helpful discussions.
One contribution of 14 to a Theme Issue ‘Topics on non-equilibrium statistical mechanics and nonlinear physics’.
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