## Abstract

Experiments in extended systems, such as the counter-rotating Couette–Taylor flow or the Taylor–Dean flow system, have shown that patterns with vanishing amplitude may exhibit periodic spatio-temporal defects for some range of control parameters. These observations could not be interpreted by the complex Ginzburg–Landau equation (CGLE) with periodic boundary conditions. We have investigated the one-dimensional CGLE with homogeneous boundary conditions. We found that, in the ‘Benjamin–Feir stable’ region, the basic wave train bifurcates to state with periodic spatio-temporal defects. The numerical results match the observations quite well. We have built a new state diagram in the parameter plane spanned by the criticality (or equivalently the linear group velocity) and the nonlinear frequency detuning.

## 1. Introduction

Progress in the understanding of spatio-temporal dynamics of extended systems far from equilibrium has been achieved during the last few decades due to model nonlinear partial differential equations (PDEs), such as the complex Ginzburg–Landau equation (CGLE) or the Kuramoto–Sivashinsky equation (Cross & Hohenberg 1993; Aranson & Kramer 2002). In many cases, for numerical simulations of these equations, periodic boundary conditions have been used (Chaté 1994; van Hecke 1998; Howard & van Hecke 2003; van Saarloos 2003). For such boundary conditions, in the case of the one-dimensional CGLE describing the pattern formation from Hopf bifurcation in systems with translational invariance, a state diagram describing the variety of regimes observed for different parameters has been established, and the stability of these states has been thoroughly investigated (Brusch *et al*. 2000). In particular, it was shown that above the Benjamin–Feir (BF) line, the pattern exhibits coherent structures, such as Nozaki–Bekki holes and spatio-temporal chaos, induced either by amplitude defects or phase defects (Chaté 1994; Montagne *et al*. 1996; van Hecke 1998; Howard & van Hecke 2003). Few experiments mimicking these conditions have been designed to validate the solutions of this equation (Kolodner *et al*. 1988). These systems are characterized by the translation invariance, and the group velocity can be eliminated from the CGLE, leading the pattern to be described by a set of two control parameters (*c*_{1}, *c*_{3}) that will be defined below. However, in a large number of experiments in bounded domains, patterns have a finite length and the perturbations decay near the lateral walls. This is the case, for example, in the rectangular Rayleigh–Bénard convection cell (Kolodner *et al*. 1986); in the pattern of thermo-capillary waves in a laterally heated liquid layer (Garnier & Chiffaudel 2001; Garnier *et al*. 2002); in spiral vortex flow in the Couette–Taylor flow between counter-rotating cylinders (Zaleski *et al*. 1985; Andereck *et al*. 1986; Tagg *et al*. 1990; Tagg 1994; Ezersky *et al*. 2004); in the travelling rolls in a cylindrical annulus with a radial temperature gradient (Lepiller *et al*. 2007); in the travelling inclined vortex flow in the Taylor–Dean system (Mutabazi *et al*. 1990; Laure & Mutabazi 1994; Bot *et al*. 1998; Bot & Mutabazi 2000); and in parametrically excited ripples in viscous fluids (Ezersky *et al*. 2001). To explain the regimes observed in these experiments on pattern formation in bounded flow systems with lateral boundaries, it is necessary to consider the conditions that are more close to experimental situations. The boundary conditions have an influence on the stability of the system and pattern generation. In fact, different results have been found from numerical resolution of the CGLE for the complex amplitude *A*(*x*,*t*) in systems with a finite length *L* by using various types of boundary conditions. Deissler (1985) has used the boundary conditions *A*_{xx}(0,*t*)=0 to mimic an open flow at the inlet *x*=0 and *A*(*L*,*t*)=0 for the outlet. Tobias & Knobloch (1998) have used the boundary conditions *A*(0,*t*)=*A*(*L*,*t*)=0 to illustrate the variety of novel behaviours that occur when unidirectional waves interact with boundaries: wall mode and front between patterns emerging from destabilization of the wave train. Tobias *et al*. (1998) have used the conditions *A*_{x}(0,*t*)=*A*(*L*,*t*)=0 to study the problem of breakup of spiral waves into chemical turbulence in which *x*=0 plays the role of the core and *x*=*L* the outer boundary. Lücke & Recktenwald (1993) have investigated the pattern growth resulting from an inlet boundary condition that is produced by thermal equilibrium transverse momentum fluctuations that are advected into the system at *x*=0. The influence of boundary conditions was also highlighted recently for a two-dimensional CGLE (Eguiluz *et al*. 1999, 2000, 2001). It was shown that homogeneous Dirichlet boundary conditions induce a strong selection mechanism of pattern wavenumber and frequency because of the absorption of disturbances on the lateral edges.

The present work deals with one-dimensional patterns described by the CGLE in finite geometry in which the amplitude of the supercritical wave pattern vanishes at the lateral boundaries of the domain. We will show numerically that, due to the group velocity, the state diagram differs fundamentally from that obtained for periodic boundary conditions (Chaté 1994; Montagne *et al*. 1996; van Hecke 1998; Brusch *et al*. 2000; Howard & van Hecke 2003; van Saarloos 2003). A sequence of periodic defects (PDs) in the BF stable region will be excited in a finite zone of the parameter space.

The paper is organized as follows: in the second section, we present the amplitude equation and the numerical algorithm; in §3, we present the results that will be discussed in §4; and §5 is devoted to the conclusion of the study.

## 2. Amplitude equation and numerical scheme

Different physical and chemical systems driven out of equilibrium may undergo Hopf bifurcations leading to a rich variety of spatio-temporal behaviours. In most cases, these bifurcations occur with broken spatial and temporal symmetries and they induce the formation of wave patterns that are described by an order parameter *A*(*x*,*t*), such that *A*=0 in the high symmetry state and *A*≠0 in the low symmetry state. The field of the wave pattern is given by Cross & Hohenberg (1993) and Aranson & Kramer (2002),(2.1)where the parameter order *A*(*x*,*t*) is a complex amplitude (*A*=|*A*|e^{iϕ}) of the slow dynamics of the waves; c.c. stands for the complex conjugate; and the eigenfunction depends on the coordinates perpendicular to the critical wavevector. When both the critical wavenumber *k*_{c} and frequency *ω*_{c} are non-zero at the pattern forming Hopf bifurcation, the primary modes are travelling wave dynamics, which can be described by the CGLE for the complex amplitude (Bot *et al*. 1998; Bot & Mutabazi 2000). In a one-dimensional system, it is usually written as(2.2)where *s* is the linear group velocity of a left travelling wave (for *s*>0); *μ* is the relative distance of the control parameter from the critical point; and the coefficients *c*_{1} and *c*_{3} describe linear dispersion and nonlinear frequency detuning, respectively. The parameters *μ* and *s* act together by means of the scaled group velocity, . In this work, we have imposed on the complex amplitude the following boundary conditions:(2.3)These boundary conditions are realized in different extended systems, where the pattern amplitude vanishes near lateral boundaries (Zaleski *et al*. 1985; Andereck *et al*. 1986; Mutabazi *et al*. 1990; Tagg *et al*. 1990; Laure & Mutabazi 1994; Tagg 1994; Bot *et al*. 1998; Bot & Mutabazi 2000; Ezersky *et al*. 2001, 2004; Garnier & Chiffaudel 2001; Garnier *et al*. 2002; Lepiller *et al*. 2007). Contrary to the problem with periodic boundary conditions, the group velocity *s* cannot be removed from the equation since the system has no Galilean invariance. Therefore, the present formulation introduces the linear group velocity as a new parameter. The length *L* of the system is fixed in the experiment, while it takes discrete values for periodic boundary conditions. Therefore, in one-dimensional systems with a finite length, the CGLE is spanned by a space of three control parameters, (*s*, *c*_{1}, *c*_{3}) while it is spanned by only two control parameters (*c*_{1}, *c*_{3}) in the case of an extended system. For convenience, we have chosen to fix the linear group velocity *s* and to vary the criticality parameter *μ*, which is more accessible in experiment than the linear group velocity. In the discussion, we will present the scaled CGLE where the parameter *μ* has been removed.

The numerical method used throughout is the finite-difference scheme in space and the fourth-order Runge–Kutta algorithm in time. In fact, the finite-difference scheme allows the discrete form of the CGLE (equation (2.2)) to be obtained. Such a discretization scheme of this equation has been used in different works, for example, for description of vortex line dynamics (Willaime *et al*. 1991) or for the determination of coefficients of the Ginzburg–Landau equation from experimental spatio-temporal data in the wakes behind an array of cylinders (Le Gal *et al*. 2003). In these two examples, the oscillation of each isolated cell (each vortex or each wake) obeys a complex Landau equation which is the normal form of the Hopf bifurcation that gives rise to each oscillator. The global behaviour of the vortex or the wake arrays was described by the dynamics of coupled Hopf oscillators. Ravoux *et al*. (2000) analysed the stability of the nonlinear plane wave solutions of the discretized CGLE. The discretized CGLE reads(2.4a)where , with 2≤*j*≤*N*−1, and the homogeneous boundary conditions (equation (2.3)) were(2.4b)Thus, we have reduced the PDE to a system of ordinary differential equations. The structure of such systems implies that their dynamics is the result of interaction between *N* individual dynamical entities. We have integrated numerically the system (2.4*a*) and (2.4*b*) using a standard fourth-order Runge–Kutta method. In our numerical simulations, we have considered a system involving *N* sites with *N*=301. The accuracy of the numerical procedure has been examined by testing different time and space steps. The time step must be small enough at a given spacing to ensure the conditional stability of the preceding algorithm at each step. Finally, the time and space increments were chosen to be Δ*t*=0.025 and Δ*x*=0.2. We have chosen as initial condition a symmetric centred pulse-like solution given by(2.5)where *A*_{0} and *γ* are constants and *x*_{a}=*L*/2 is the position of the maximum of this initial perturbation. Since the initial perturbation must satisfy the boundary conditions (2.4*b*), *A*_{0} must be a small quantity and *γ* a large one. That is why, in our numerical calculations, we have chosen *A*_{0}=0.01 and *γ*=100. Thus, depending on the nature of the nonlinear dynamics of the system, the initial disturbance can grow and invade the domain (global mode) or it can be expelled from the system (convective instability).

## 3. Results of numerical simulations

The control parameters *c*_{1}, *s* and *L* were fixed at *c*_{1}=0.50, *s*=0.50 and *L*=60, and we have varied *μ* and *c*_{3}. This variation enabled us to identify, in the parameter space (*μ*, *c*_{3}), zones in which the patterns exhibit different behaviours.

A linear growth analysis of the CGLE (2.2) in unbounded domains along the lines of Cross & Hohenberg (1993) shows that the far front separating the base state (*A*=0) and the bifurcated state (*A*≠0) moves with velocity . In the convectively unstable region 0<*μ*<*μ*_{a}, the front propagates upstream so that *V*_{f}<0, the scaled group velocity *V*_{g}>2 and the amplitude of the pattern decays locally (figure 1*a*). In the absolutely unstable regime, *μ*>*μ*_{a}, it moves downstream, i.e. *V*_{f}>0, the scaled group velocity *V*_{g}<2 and the bifurcated state (global mode) invades the whole system (figure 1*b*). At the boundary *μ*=*μ*_{a}, the front is stationary and the scaled group velocity *V*_{g}=2. To validate our numerical scheme, we have retrieved the analytical results for absolute and convective instability for our parameters *s*, *c*_{1} and *L*. For the chosen values of these parameters, the transition between convective instability and absolute instability occurs at *μ*_{abs}=0.05, i.e. *V*_{f}=0 (Tobias *et al*. 1998). Moreover, for *μ*=0.065, the velocity of the front propagation found numerically is *V*_{f}≈0.08 for *L*=60 (figure 1*c*), which is in a good agreement with the analytical *V*_{f}≈0.07 in the limit *L*→∞. With increasing *μ*, the nonlinear wave train broadens out to a larger value of *x*, until it fills the entire domain.

Our goal was to investigate the effects of the nonlinear dispersion coefficient *c*_{3} of the CGLE on the dynamics of the system, with the prescribed boundary conditions (2.3). Solving the CGLE for several values of *c*_{3}, we found that, in the convective instability regime, the system is practically insensitive to the variation of the coefficient of the nonlinear dispersion. In the absolute instability regime (i.e. for *μ*≥0.05), the variation of the coefficient *c*_{3} leads to bifurcation of the global mode to new states that are summarized in the state diagram of figure 2 for *c*_{3}∈[−5;5]. The global mode is stable between lines *L*_{3}−*L*_{2} and *L*_{1}. Above the line *L*_{1} and below the line *L*_{2}, the global mode pertains to a secondary instability that leads to different types of pattern, depending on the sign of *c*_{3} and the value of *μ*.

The space–time diagram of the phase of the global mode near the line *L*_{1} (figure 3) shows that the pattern propagates in only one direction, with an average frequency *ω*=0.06 and wavenumber *k*=0.30. The behaviour of the global mode above the line *L*_{1} is represented by figure 4 for *μ*=0.15 and *c*_{3}=3.65. The regular pattern is destabilized and gives rise to a modulated state in space and time near the wall *x*=0, while, near the end *x*=*L*, the pattern remains non-modulated. We therefore have obtained a state with two patterns separated by a front at the position *x*_{f}. Figure 4*a*,*b* shows a regular wave train in *x*_{f}<*x*<*L* and a modulated pattern in 0<*x*<*x*_{f}, with wavenumber *k*_{m}≈1.07 and frequency *ω*_{m}≈1.40. The front position moves in time with the group velocity *s*. It moves towards the wall *x*=*L* with increasing *c*_{3}.

For negative values of *c*_{3}, the global mode is destabilized differently. In the parameter region between *L*_{3} and *L*_{4}, the global mode exhibits amplitude defects: points in the space–time diagram where the amplitude of the wave vanishes (|*A*|≈0) and the phase (*ϕ*=arg(*A*)) is not defined (figure 5). These amplitude defects appear at regular time intervals; for this reason we have called them *time-PDs*, and the frequency of their appearance increases with increasing values of *μ* and *c*_{3}. On the space–time diagram of figure 5*a*, one observes a sequence of four-point defects on an interval of time *t*≈100. The slope of the defects observed in figure 5*a* indicates the direction of wave propagation.

In order to characterize these amplitude defects, we have computed the hydrodynamic field *u*(*x*, *t*) defining the wave packet in the simplest form,(3.1)where c.c. stands for the complex conjugate term. In our numerical simulations, we have fixed *ω*_{0}=1.5 and *k*_{0}=2.0, taken from the experimental data of the Couette–Taylor flow (Ezersky *et al*. 2004). The space–time diagram of figure 5*b* shows the grey level of the phase of the field *u*(*x*,*t*). The phase varies from −*π* (black) to +*π* (white) and undergoes a jump of *π* in the position of the core of a defect. The space–time diagram of figure 5*c* is that of the phase perturbation obtained after a complex demodulation. The amplitude profile of the global mode in the neighbourhood of a single defect is shown in figure 6*a*. The core of the defect is located at *x*_{d}≈15. Figure 6*b* illustrates the temporal evolution of the mode amplitude along the four defects.

In the vicinity of an isolated defect, levels of constant amplitude are ellipses stretched along one direction. Figure 7*a* shows the defect situated at the position . The angle of inclination of each ellipse is approximately the same for all point defects. We have verified that the inclination of the ellipse is determined by the group velocity of the wave pattern. For example, the inclination of the ellipse axis in figure 7*a* is 0.6, while the group velocity of perturbations is *s*=0.5. It should be emphasized that, unlike Nozaki–Bekki holes which move with definite velocity, amplitude defects exist in the space–time diagram at a definite time and coordinate. Nevertheless, the group velocity of the patterns is displayed by the distribution of amplitude and phase in the vicinity of the defect. The phase of the wave pattern in the vicinity of the point defect is shown in figure 7*b*, and the clockwise circulation of the gradient phase around this point is . In the vicinity of an isolated defect, the amplitude and phase fields can be represented by a simple model of complex amplitude,(3.2)where *Χ*=*x*−*x*_{d}; *τ*=*t*−*t*_{d}; and *β* is an ad hoc parameter. It is possible to check that such a function is a solution of equation (2.2). The levels of constant amplitude and constant phase are topologically similar to those obtained numerically in the vicinity of the defects. The similarity includes the inclination of ellipses (lines of constant amplitude in figure 7*c*,*d*), which are related to the group velocity.

The position of the defects in the pattern depends on the values of the control parameters (*μ*, *c*_{3}). For small values of *μ*, defects appear in the vicinity of the wall (*x*=0). When the parameters *μ* and *c*_{3} are increased, defects move towards the interior of the domain. Figure 8*a* presents the space–time diagram of the amplitude of the wave pattern in the zone between the lines *L*_{2} and *L*_{4} in figure 2. We have selected the values of parameters *μ* and *c*_{3} close to the line *L*_{4}. The sequence of PDs appears now in the middle of the domain and travelling holes-like solutions are observed near the wall *x*=0. Figure 8*b* is the space–time diagram of the phase of the constructed pattern field. When one moves away from the line *L*_{4}, the behaviour of the pattern becomes more complex with irregular distribution of a large number of holes and amplitude defects.

## 4. Discussion

In the simulations of the CGLEs with homogeneous boundary conditions, we have found new states that were different from those previously obtained by many authors that used periodic boundary conditions (Chaté 1994; Montagne *et al*. 1996; van Hecke 1998; Brusch *et al*. 2000; Howard & van Hecke 2003; van Saarloos 2003) or homogeneous boundary conditions (Tobias *et al*. 1998). The importance of boundary conditions on solutions of a PDE is a well-known property (Tikhonov & Samarskii 1990). Tobias & Knobloch (1998), Tobias *et al*. (1998) and Eguiluz *et al*. (1999, 2000, 2001) have shown that homogeneous boundary conditions may change significantly the pattern generated in the system. In the case of positive values of *c*_{3}, we have found states (figures 3 and 4) that are similar to those obtained by Tobias *et al*. (1998). These authors have investigated thoroughly the stability of the obtained states, and have shown that the transition is ruled by the change of the nature of the instability of the basic wave train from convective to absolute secondary instability. For negative values of *c*_{3} (the BF stable region), we have found new patterns that exhibited temporal periodic amplitude defects with or without travelling holes, depending on the values of *c*_{3}. The stability analysis of the basic wave train for *c*_{3}<0 is beyond the scope of the present work. To our best knowledge, these states were not reported in numerical simulations. In fact, many previous studies (Chaté 1994; Montagne *et al*. 1996; van Hecke 1998; Brusch *et al*. 2000; Howard & van Hecke 2003; van Saarloos 2003) of the CGLE with periodic boundary conditions reported amplitude defect generation in the region of BF instability, i.e. when 1−*c*_{1}*c*_{3}<0. In our case, since we have fixed *c*_{1}=0.5, the BF unstable zone corresponds to the region with *c*_{3}>2 (figure 2). Temporal periodic amplitude defects were obtained for *c*_{3}<0. Such defects have been observed in the spiral pattern in the counter-rotating Couette–Taylor system (Ezersky *et al*. 2004), in the travelling roll pattern in the cylindrical annulus with a radial temperature gradient (Lepiller *et al*. 2007), in the Taylor–Dean system (Mutabazi *et al*. 1990; Bot *et al*. 1998; Bot & Mutabazi 2000) or in binary mixture convection (Voss *et al*. 1999). Appearance of periodic sequences of point defects in spatio-temporal diagrams seems to be typical for the systems with rotation symmetry (Ostrovsky & Potapov 1999). Numerical simulations of the anisotropic CGLE have presented a class of solutions where the defects were aligned spontaneously along chains (Weber *et al*. 1991). These chains, which bear some resemblance with chevron patterns, have been observed in liquid crystal convection (Rossberg & Kramer 1998) and have been discussed in more detail by Weber *et al*. (1992) and Faller & Kramer (1999).

We should emphasize that periodic amplitude defects can be obtained only for homogeneous boundary conditions, and cannot be found for periodic boundary conditions. This may be explained as follows: for solutions with defects, the phase of the complex amplitude *A*(*x*, *t*) at a given time *t* is not periodic andIt is impossible to glue this solution at the ends in order to satisfy periodic boundary conditions. The length *L* of the domain plays a crucial role in our study. We have varied the length *L* and found that the observed secondary structures appeared at positions *x*_{f} or *x*_{d}, which increases with the length *L*. The position of the defects in the pattern depends on the values of the control parameters (*μ*, *c*_{3}), but also on the initial conditions; for different initial conditions, PDs appeared at different coordinates *x*_{d}.

In order to exclude the influence of the parameter *μ* and for a better comparison with previous results, we have solved the CGLE in the scaled form(4.1)where we have introduced the scaled coordinate *X*=*μ*^{1/2}*x* and time *T*=*μt*. The new scaled group velocity *V*=*sμ*^{−1/2} and the domain length becomes *l*=*μ*^{1/2}*L*. The corresponding state diagram has been plotted in figure 9 in the parameter plane (*V*,*c*_{3}) for *l*=600. One can observe that the state diagram in the parameter plane (*V*,*c*_{3}) of equation (4.1) is almost equivalent to that in the parameter plane (*μ*, *c*_{3}) of equation (2.2) presented in figure 2. The difference occurs for small values of the linear group velocity (*V*→0). When the linear group velocity vanishes (*V*=0) and *c*_{1}=0.50 and *c*_{3}=1.76, we retrieved the spatio-temporal intermittency of van Hecke (1998). We have plotted in figure 10*a* the characteristic pattern of a spatio-temporal intermittent regime for *V*=0.05, in which a global mode coexists with a chaotic attractor: the state consists of patches of plane waves, separated by various holes, i.e. local structures characterized by depression of the amplitude |*A*| of the pattern. Similar patterns called ‘hole-defect chaos’ were obtained in simulations of equation (4.1) with periodic boundary conditions (Howard & van Hecke 2003) for parameters *c*_{1}=0.6, *c*_{3}=1.4 and *l*=500. Our simulations show that, for a small group velocity, homogeneous boundary conditions only slightly modify the region of ‘hole-defect chaos’.

## 5. Conclusion

We have performed a numerical study of the CGLE in a one-dimensional extended system with homogeneous boundary conditions. We have fixed the domain length and the linear dispersion coefficient, while we varied the criticality or, equivalently, the group velocity and the nonlinear dispersion coefficient. We have shown that, depending on the sign of the nonlinear dispersion coefficient, secondary structures were excited from a regular wave pattern, in the form of amplitude modulated waves or periodic in time amplitude defects. These periodic amplitude defects have been observed in many experiments exhibiting patterns with vanishing amplitude at lateral boundaries, such as patterns of Taylor spirals or Taylor–Dean rolls. The stability analysis of the basic wave train for negative nonlinear frequency detuning is an open question. The numerical investigation of the CGLE with homogeneous boundary conditions for different values of the control parameters (group velocity; criticality parameter; linear and nonlinear dispersion coefficients) represents a big issue in the understanding of many physical systems with pattern formation.

## Acknowledgments

L.N. would like to thank the Agence Universitaire de la Francophonie (AUF) for sponsoring his research stay to the LOMC. Part of this work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. Financial support from the Swedish International Development Agency (SIDA) is acknowledged.

## Footnotes

- †
↵† Permanent address: Département de Physique, Faculté des Sciences, Université de Douala, PO Box 24157 Douala, Cameroun.

- ‡
↵‡ Permanent address: Laboratoire de Morphodynamique Continentale et Côtière, UMR 6143 CNRS-Université de Caen-Basse Normandie, 14000 Caen, France.

- Received January 2, 2009.
- Accepted April 1, 2009.

- © 2009 The Royal Society