Phase-Diffusion Equations for the Anisotropic Complex Ginzburg-Landau Equation
Derek Handwerk, Gerhard Dangelmayr, Iuliana Oprea, Patrick D. Shipman
PPhase-Diffusion Equations for the Anisotropic ComplexGinzburg-Landau Equation
Derek Handwerk, Gerhard Dangelmayr, Iuliana Oprea, and Patrick D. Shipman
Department of Mathematics, Colorado State University, Ft. Collins, CO 80523-1874September 29, 2020
Abstract
The anisotropic complex Ginzburg-Landau equation (ACGLE) describes slow modulations ofpatterns in anisotropic spatially extended systems near oscillatory (Hopf) instabilities with zerowavenumbers. Traveling wave solutions to the ACGLE become unstable near Benjamin-Feir-Newell instabilities. We determine two instability conditions in parameter space and studycodimension-one (-two) bifurcations that occur if one (two) of the conditions is (are) met. Wederive anisotropic Kuramoto-Sivashinsky-type equations that govern the phase of the complexsolutions to the ACGLE and generate solutions to the ACGLE from solutions of the phaseequations.
Key words:
Anisotropy, Ginzburg-Landau equation, phase equation, Kuramoto-Sivashinsky equa-tion, Benjamin-Feir-Newell instability.
Complex spatiotemporal patterns such as spatiotemporal chaos and defects, manifest themselvesin spatially extended systems driven far from equilibrium. The problem of finding a general frame-work for the characterization of such disordered states, as well as the identification of instabilitymechanisms generating them, remain active areas of investigation in nonlinear dynamics. Whilecomplex spatiotemporal patterns in nonequilibrium isotropic systems have been intensively studied,far less is known about complex spatiotemporal dynamics in anisotropic systems. Yet, intriguingfeatures of patterns in anisotropic media when driven out of equilibrium have been observed in awide range of experimental studies. Examples include electroconvection of nematic liquid crystals[24], surface nanopatterning by ion-beam erosion [33], chemical waves in catalytic surface reactions[32, 34], epitaxial growth [35], sea ice melting [4], and vegetation patterns [7].These experimental observations and numerical simulations demonstrate that anisotropy canlead to novel mechanisms and phenomena that are manifested only in anisotropic media. Forinstance, Rayleigh-B´enard convection [3, 13], a prototype of spatiotemporal chaos in isotropic fluids,displays a paradigm of spatiotemporal chaos known as spiral defect chaos, while in anisotropicsystems such as nematic electroconvection other mechanisms intervene, leading to new patternssuch as the zigzag spatiotemporal chaos [11]. Similar patterns can arise from even this wide rangeunderlying physical systems. Analysis of mathematical models for these diverse systems results ina universal characterization of the similar patterns by their description using amplitude and phase equations. 1 a r X i v : . [ n li n . PS ] S e p mplitude equations describe the slow modulation of the pattern near the threshold of instabil-ity. They can be derived via symmetry arguments or through multiple-scales analysis. The sameamplitude equation can be derived from different underlying equations up to some unknown coeffi-cients that determine the length and time scales, and the effect of the nonlinearity. The underlyingequations can be used to determine the correct coefficients of the amplitude equation or they canbe scaled out completely. The ability to completely remove the system-dependent coefficients fromthe equation demonstrates the universality of amplitude equations.The slow modulations of a complex amplitude satisfying an amplitude equation are generallygoverned by phase equations which describe the extremely slow variation of the phase of the ampli-tude. When a system has more than one extended spatial direction, it and thus also the amplitudeand phase equations, can be either isotropic or anisotropic. Nematic liquid crystals [9, 10, 24], ionbombardment [19], and surface erosion and growth [35] are examples of physical scenarios that canbe described by anisotropic amplitude and phase equations.The amplitude equation of concern in this paper is the two-dimensional, anisotropic complexGinzburg-Landau equation (ACGLE), ∂ t A = µA + (1 + iα ) ∂ x A + (1 + iα ) ∂ y A − (1 + iβ ) | A | A, (1)where A is the complex amplitude and α , α , β, µ ∈ R with µ >
0. For α = α , this equation isthe isotropic complex Ginzburg-Landau equation (CGLE). We refer to (1) as the 1D CGLE if the y -dependent term is absent and one looks only for solutions A ( x, t ).Since the spatially independent part of the complex Ginzburg Landau equation coincides withthe normal form for a supercritical Hopf bifurcation [16], both the 1D and 2D isotropic complexGinzburg Landau equation and variants thereof, referred to as λ − ω systems as incepted by Kopelland Howard [23], have been studied as spatiotemporal model equations showing plane wave andfront solutions in various settings including reaction diffusion systems [20, 25, 26, 39] as well aspredator-prey systems [2, 37]. Extending these studies for the ACGLE (1) should provide newinsights into the effect of ansiotropies on this kind of spatiotemporal dynamics.There has been much less research into the ACGLE compared to the isotropic CGLE. Someaspects of phase chaos were investigated in [14], and new chevron-like, ordered defect solutionswere reported in [15]. A study of a perturbed amplitude in the longwave case was performed in [5].The authors of this paper used a perturbed ansatz of the form A ( x, y, t ) = √ µ (1 + r ( x, y, t )) e − iβµt that lacks any phase perturbation as considered by us (see Equation (21)), which is crucial forthe reduction of (1) to a phase equation. A comprehensive analysis of long-wave and short-waveinstabilities of traveling wave solutions of the ACGLE was performed in [10]. This analysis was donein the context of modulational instabilities of traveling waves determined as solutions of systemsof two or four globally coupled complex Ginzburg-Landau equations, which are the amplitudeequations corresponding to oscillatory instabilities with nonzero critical wavenumbers of a basicstate of an anisotropic system [36].Due to the assumptions used in the derivation of phase equations from amplitude equations, it isimportant to consider the regions and circumstances in which they are valid. In the case of the 1DCGLE, the validity of phase equations has been established in [17, 31, 40] for different parameterregimes. In the parameter regime where a 1D, periodic traveling wave solution A ( x, t ) = A q e i ( qx + ω q t ) is stable to perturbations (Eckhaus stable) the validity has been looked at in [31], while near theEckhaus instability the validity of a Korteweg de Vries equation was proved in [17]. Near the so-called Benjamin-Feir-Newell instability, where all traveling wave solutions are unstable, a fourth-2rder diffusive equation called the (1D) Kuramoto-Sivashinsky (KS) equation [27, 38] has beenestablished [40].A derivation and partial analysis of the 2D, isotropic extension of the 1D Kuramoto-Sivashinsky(KS) equation as the equation for the phase dynamics of the (2D) CGLE near the Benjamin-Feir-Newell instability is given in [27]. The form of this equation is ∂ T Φ = − ( ∂ X Φ + ∂ Y Φ) − ( ∂ X Φ + 2 ∂ X ∂ Y Φ + ∂ Y Φ) + 12 (cid:0) ( ∂ X Φ) + ( ∂ Y Φ) (cid:1) , (2)where ( X, Y, T ) are slow variables. This equation has been studied in its own right as a model forphenomena such as flame fronts and the leading edge of a viscous fluid flowing down an inclinedplane as well as for its spatiotemporal chaotic behavior [25, 27, 38]. If all Y -dependent terms areabsent, it reduces to the 1D KS equation.In this paper we establish anisotropic versions of the KS equation as well as another 2D exten-sion of the 1D KS equation as equations governing the phase dynamics of the ACGLE near differentBenjamin-Feir-Newell-type instabilities. The validity of these equations is confirmed through nu-merical simulations. Beyond its relevance for the ACGLE, an anisotropic 2D KS equation has beenbeen introduced as a model for surface sputter erosion and epitaxial growth [35]. For an in-depthnumerical study of (2), we refer to [21]. In this section we study the linear stability of traveling plane-wave solutions (TPWS’s) of theACGLE (1). Note that the anisotropy presents itself in the linear dispersion terms. A TPWS tothe ACGLE in the k -direction and with frequency ω is given by A = R e i ( k · x − ωt ) , (3a) R = µ − ( k + k ) , (3b) ω = βR + α k + α k , (3c)where k = ( k , k ) is the wavenumber and x = ( x, y ). The requirements (3b) and (3c) are foundby substituting (3a) into (1).To determine the stability of a TPWS (3) we analyze the time-evolution of perturbations.Separating the perturbations r ( x, y, t ) of the amplitude and φ ( x, y, t ) of the phase as A ( x, y, t ) = R (1 + r ( x, y, t )) e i [ k · x − ωt + φ ( x,y,t )] , (4)substituting (4) into (1), and then separating real and imaginary parts leads to the evolutionequations ∂ t r = − R r − R r − R r + ∂ x r + ∂ y r − ( ∂ x φ ) − ( ∂ y φ ) − r ( ∂ x φ ) − r ( ∂ y φ ) − k α ∂ x r − k α ∂ y r − k ∂ x φ − k ∂ y φ − k r∂ x φ − k r∂ y φ − α ∂ x φ − rα ∂ x φ − α ∂ y φ − rα ∂ y φ − α ∂ x r∂ x φ − α ∂ y r∂ y φ, (5)3nd ∂ t φ = − R βr − R βr − α ( ∂ x φ ) − α ( ∂ y φ ) + ∂ x φ + ∂ y φ − k α ∂ x φ − k α ∂ y φ + 11 + r (cid:20) α ∂ x r + α ∂ y r + 2 k ∂ x r + 2 k ∂ y r + 2 ∂ x r∂ x φ + 2 ∂ y r∂ y φ (cid:21) . (6)for the perturbations. For details of the computation of (5) and (6) we refer to [18].Upon linearizing around the base state ( r, φ ) = (0 ,
0) of the unperturbed TPWS, (5) and (6)simplify to ∂ t r = − R r + ∂ x r + ∂ y r − k α ∂ x r − k α ∂ y r − k ∂ x φ − k ∂ Y φ − α ∂ X φ − α ∂ Y φ, (7) ∂ t φ = − R βr + ∂ x φ + ∂ y φ − k α ∂ x φ − k α ∂ y φ + α ∂ x r + α ∂ y r + 2 k ∂ x r + 2 k ∂ y r. (8)Substituting the modes r = ˆ re σt + i q · x and φ = ˆ φe σt + i q · x with wavevector q = ( q , q ) into (7) and(8) leads to an eigenvalue problem for (ˆ r, ˆ φ ) with eigenvalue σ . The trace and determinant of thecorresponding matrix M areTr M = − R + | q | ) − i ( α k q + α k q ) , (9)det M = 2 q (cid:0) R (1 + α β ) − α ) k (cid:1) + 2 q (cid:0) R (1 + α β ) − α ) k (cid:1) − α α ) k k q q + | q | + ( α q + α q ) (10)+ 4 iR (cid:0) ( α − β ) k q + ( α − β ) k q (cid:1) + 4 i ( α − α ) q q ( k q − k q ) . We are primarily interested in the stability of TPWS’s against long-wavelength (LW) perturbationswhere | q | is arbitrarily small, 0 < | q | (cid:28)
1. Expanding the two roots of the characteristic equation σ − σ Tr M + det M = 0 for small ( q , q ) yields one eigenvalue σ s = − R + O ( | q | ) with negativereal part for sufficiently small | q | , and another eigenvalue that determines the stability of the TPWSagainst LW perturbations, σ ( q ; k ) = 2 ik ( β − α ) q + 2 ik ( β − α ) q + (cid:16) k R + 2 β k R − α β − (cid:17) q + (cid:16) k k R + 4 β k k R (cid:17) q q + (cid:16) k R + 2 β k R − α β − (cid:17) q + O ( | q | ) . (11)The real part of this eigenvalue can be written as σ r ( q ; k ) = Q ( q ; k ) + O ( | q | ) , (12)where Q ( q ; k ) is the following quadratic form with respect to q , with k considered as a parameter; Q ( q ; k ) = D xx ( k ) q + 2 D xy ( k ) q q + D yy ( k ) q , (13)with D xx = 2 k (1 + β ) R − ( α β + 1) , D xy = 2 k k (1 + β ) R , D yy = 2 k (1 + β ) R − ( α β + 1) . (14)4hese coefficients are in agreement with the linear stability analysis of the isotropic CGLE [29].Note that for fixed k , Q ( q ; k ) is the quadratic form with respect to q that is associated with thesymmetric matrix D ( k ) = (cid:18) D xx ( k ) D xy ( k ) D xy ( k ) D yy ( k ) (cid:19) . (15)We introduce the following notions of LW-stability and -instability. Definition 1.
A TPWS with wavenumber k is(i) LW-stable if σ r ( q ; k ) < q ,(ii) partly LW-unstable if there exist arbitrarily small, nonzero wavenumbers q , ˜ q such that σ r ( q ; k ) < σ r ( ˜ q ; k ) > σ r ( q ; k ) > q ,(iv) LW-unstable if (ii) or (iii) hold.Since σ r ( ; k ) = 0, property (i) is satisfied if q = (0 ,
0) is a strict local maximum of σ r ( q ; k ) forfixed k , which is the case if D ( k ) is negative definite. Similarly, property (iii) is satisfied if q = (0 , σ r ( q ; k ) for fixed k , which is the case D ( k ) is negative definite. D ( k ) isnegative (positive) definite if and only if its determinant is positive and one of the diagonal entries D xx , D yy is negative (positive). A sufficient condition for (ii) is that the determinant of D ( k )be negative, in which which case Q ( q ; k ) defines a saddle surface so that there are regions in the( q , q )-plane in which Q ( q ; k ) > Q ( q ; k ) <
0. Thus, to analyze the stability properties of agiven TPWS, we have to take the determinantdet D ( k ) = ( α β + 1)( α β + 1) − β ) R (cid:16) ( α β + 1) k + ( α β + 1) k ) (cid:17) (16)into consideration.The stability properties (i)-(iii) occur in the following parameter regimes: Theorem 1.
Suppose that (1 + α β )(1 + α β ) (cid:54) = 0 and let k be the wavenumber of a TPWS suchthat R > . Define F ( k ) by F ( k ) = (cid:32) β )1 + α β (cid:33) k + (cid:32) β )1 + α β (cid:33) k , (17) and assume that the coefficients of k and k are both nonzero.(a) Suppose α β > and α β > . Then F ( k ) = µ defines an ellipse that is inscribed inthe µ -circle | k | = µ . If k is in the interior of that ellipse the TPWS is LW-stable while for k outside of the ellipse the TPWS is partly LW-unstable.(b) Suppose α β < and α β > . Then the curve F ( k ) = µ is either a hyperbola or anellipse. In either case this curve intersects the µ -circle in the four points defined by k = − (1 + α β ) µ ( α − α ) β , k = (1 + α β ) µ ( α − α ) β . (18) Moreover, the TPWS is partly unstable if F ( k ) < µ and fully unstable if F ( k ) > µ .(c) Suppose α β < and α β < . Then every TPWS is fully unstable. roof. To simplify notation we set ˜ α j = 1 + α j β , j = 1 ,
2. When multiplying the expression (16)for det D ( k ) by R / ( ˜ α ˜ α ) and substituting R = µ − k − k one can see directly that(i) if ˜ α ˜ α >
0, then det D ( k ) > µ > F ( k ), and det D ( k ) < µ < F ( k );(ii) if ˜ α ˜ α <
0, then det D ( k ) > µ < F ( k ), and det D ( k ) < µ > F ( k ).Part (a) then follows directly from (i) as for ˜ α , ˜ α both positive we have det D ( k ) > µ = F ( k ), which makes D ( k ) negative definite, and det D ( k ) < µ -circle is clear as the two coefficients of F ( k ) are both > α , ˜ α both negative we have µ > k + k > F ( k ), hencedet D ( k ) is always positive. Since ˜ α < D xx ( k ) > D ( k ) is positive definite if ˜ α , ˜ α areboth negative.Lastly, part (b) follows from (ii) in the same way as (a) follows from (i). Computing theintersections of the curve µ = F ( k ) with the circle µ = | k | is straightforward.The case distinctions in Definition 1 and Theorem 1 are formulated with strict inequalities andso do not include borderline cases. We treat borderline cases such as 1 + α β = 0 or F ( k ) = µ asboundary sets of positive codimension in parameter space. Case (b) of Theorem 1 actually coverstwo cases, with the second case obtained via the parameter swap ( α , k , q ) ↔ ( α , k , q ).In Figure 1a-d we illustrate case (b) of Theorem 1. The division of the ( k , k )-plane is depictedin a and b for the cases when F ( k ) = µ defines hyperbolic and elliptic curves, respectively. InFigure 1c, d we show the division of the ( q , q )-plane into regions with Q > Q < k = (0 ,
0) and k = ( k ∗ , k ∗ ), which is close to the F ( k ) = µ curve, marked in Figure1a. When k approaches this curve the two lines separating the four regions merge and the region Q < D ( k ) → Eckhaus stability boundary
The ellipse F ( k ) = µ in case (a) of Theorem 1 extends theExkhaus stability boundary for the CGLE in the 1D and 2D cases (see, e.g. [1, 8]) to the 2Danisotropic case of the ACGLE. In particular, in the 2D isotropic case ( α = α ) the ellipse becomesa circle. The elliptic stability boundary along with the conditions 1 + α j β > j = 1 ,
2, for theACGLE was already established in [10] in a more general setting and using a different notation. In[10], in addition to the LW-stability boundary, short-wavelength instabilities have been analyzedthat may preceed the LW-instability when | k | is increased along a ray emanating from the origin,thereby extending the stability analysis pursued in [41] for the 1D case. Benjamin-Feir-Newell stability boundary
The stability analysis in [10] is exclusively for thecase when stable TPWS’s exist, that is, when 1 + α β > α β >
0. These two conditionsextend the Newell criterion [28], 1 + αβ >
0, for the existence of stable TPWS’s of the CGLE,when α = α ≡ α , to the anisotropic case. For the CGLE, the curve in the ( α, β )-plane definedby 1 + αβ = 0 is referred to as the Benjamin-Feir-Newell- (BFN-)stability boundary [1, 6]. Acomprehensive numerical study of the CGLE in the BFN-unstable as well as BFN-stable-regimeswas performed in [6]. 6 a) (b)(c) (d) Figure 1: (a) Circle | k | = µ = 1 (dashed) and the segments of the curve F ( k ) = 1 (solid) thatare inside that circle for α = − . α = 2, β = 1 .
1. For these parameters F ( k ) = µ defineshyperbolae. The regions in the ( q , q )-plane where Q > Q < k , k ) = (0 ,
0) and ( k ∗ , k ∗ ) = (0 . , . α = − α = 8, β = 1 .
1; here F ( k ) = µ defines an ellipse.In the anisotropic case of the ACGLE (1), the boundary separating the parameter region with nostable TPWS’s from the region in which stable TPWS’s exist is given by the condition 1 + α β = 0or 1 + α β = 0. This defines two surfaces in the ( α , α , β )-space that intersect in the plane α = α in the BFN-stability boundary for the CGLE. These two surfaces, which we also refer toas the BFN-stability boundary, along with the regions from Theorem 1, are visualized in Figure2. In Section 3 we will study phase equations governing the evolution of solutions A ( x, y, t ) to theACGLE for parameters near near the BFN-stability boundary. Linear and nonlinear phase equations
The eigenvalue (11) governs the linearization of theevolution equation for the spatial Fourier transform ˆ φ , with wavenumbers q = ( q , q ), of the phase φ with ∂ t ˆ φ = σ ( q ; k ) ˆ φ . Using the long-wave base-state k = k = 0 (bulk oscillation, for this state σ has no odd powers of q ), truncating the expansion of σ at fourth order, and then taking theinverse Fourier transform of the truncated equation for ˆ φ yields the linear phase equation ∂ t φ = (1 + α β ) ∂ x φ + (1 + α β ) ∂ y φ − α (1 + β )2 R ∂ x φ − α (1 + β )2 R ∂ y φ − α α (1 + β ) R ∂ x ∂ y φ. (19)7igure 2: Separation surfaces 1 + α β = 0, 1 + α β = 0 for β > ) and (b ) distinguish 1 + α β < α β <
0, respectively.This equation can be extended to a nonlinear equation for φ using the facts that (since k = k = 0)the equation must be invariant under x → − x and y → − y , and that it depends only on spatialderivatives. The lowest-order nonlinear terms that satisfy these conditions are ( ∂ x φ ) and ( ∂ y φ ) .Adding these terms to (19), we arrive at ∂ t φ = (1 + α β ) ∂ x φ + (1 + α β ) ∂ y φ − α (1 + β )2 R ∂ x φ − α (1 + β )2 R ∂ y φ − α α (1 + β ) R ∂ x ∂ y φ + g ( ∂ x φ ) + h ( ∂ y φ ) , (20)with yet unknown coefficients g and h . From the phase equation (20), we see that the BFN-instability 1 + αβ < α i values because of the anisotropy in the ACGLE (1). This again demonstrates thepossibility of a traveling plane wave to be stable in one direction and unstable in the other givingrise to chaotic solutions that do not occur for the isotropic CGLE.In Section 3 we derive a phase equation of the form of (20) for slow variables using a multiple-scale expansion. Examples for the effect of anisotropy on chaotic solutions
As pointed out previously, thesolutions of the ACGLE encompass those of the CGLE (for α = α ), but the added anisotropyallows for solutions that are not possible in the isotropic case. Not only can traveling plane waveshave different degrees of stability or instability in the x and y directions, they can now be bothstable and unstable to perturbations depending on direction, as captured by Theorem 1(b) andillustrated in Figure 1. For example, in the CGLE phase chaos demonstrates itself as an evolvingcellular structure [6]. This type of phase chaos can exist in the ACGLE, but as one of the lineardispersion parameters, say α , is adjusted so that, as traveling waves become BFN stable in onedirection, the isotropic cellular behavior gives way to a phase-chaotic structure of ripples which8igure 3: Snapshots of chaotic solutions to the ACGLE, where traveling plane waves are stable inthe y direction and unstable in the x direction. Both figures show | A | . (a) Time snapshot of asimulation with parameter values α = − . , α = 2, β = 1 .
1, at time t = 600. This simulationshows phase chaos where the amplitude is bounded away from zero, i.e. | A | >
0, and is seento be around 1. The cellular structure of the phase chaotic solution of the CGLE, see [6], hasbeen replaced with a ripple structure. (b) Time snapshot of a simulation with parameter values α = − , α = 2, β = 1 .
1, at time t = 600. The stability in the y direction is also apparent in thisdefect-chaos parameter regime, which shows partial coherence in the stable direction. The regionsof defects appear to travel along the unstable horizontal direction.are aligned along the stable direction; see Figure 3a (all numerical solutions shown in this paperare for µ = 1). A similar behavior also happens for parameter values which yield defect chaos inthe CGLE. As one direction is made BFN-stable, ripples appear aligned along the stable direction.Defects appear and travel mostly along the unstable direction, see Figure 3b.Figure 4 illustrates the effect of different linear dispersion coefficients on hole and shock wallsolutions of the isotropic CGLE. In this section, we first establish a general phase equation for the ACGLE (1) by applying amultiple-scale expansion to the perturbational amplitude r and phase φ in (4) for k = k = 0. Theresulting system of equations allows the amplitude to be slaved to the phase, leaving an equation forthe phase alone. In this derivation, no assumptions are yet made about the parameters. We thenspecify this phase equation for the case when ( α , α , β ) is close to the BFN-stability boundary, forwhich we distinguish two cases: the case 1 + α β = 0 and 1 + α β > α β = 0 and 1 + α β = 0 (codimension-two, Subsection 3.2).Substituting k = (0 ,
0) into the traveling plane wave solution (3) leads to the following ansatz9igure 4: Four snapshots of solutions to the ACGLE (1). a) Hole and shock-wall solution. Theshock walls keep the spiral defects separated. Here α = α = − .
22. b) The shock walls no longercontain the spirals which are free to diffuse and annihilate. Here α = α = 0 .
22. c) Spiral-defectchaos with α = − · .
22 and α = − .
22. d) The previous 3 solution types can occur in theisotropic case, but this is a uniquely anisotropic solution. There are spiral defects, whose centersare skewed into ellipses by the anisotropy, together with phase-chaotic ripples in the y -direction.All solutions have β = 1 . t = 600 on a grid of [ − , × [ − , − , × [ − , A with perturbed amplitude and phase, A ( x, y, t ) = √ µ (1 + r ( x, y, t )) e i ( − βµt + φ ( x,y,t )) . (21)For the slow space and time scalings, we use r ( x, y, t ) = δ W ( δx, δy, δ t ) , φ ( x, y, t ) = δ Φ( δx, δy, δ t ) , (22)extending the scaling introduced in [12] for the 1D CGLE to the ACGLE. After separating realand imaginary parts, scaling, and dividing the W and Φ equations each by δ , we obtain δ ∂ T W = δ ( ∂ X W + ∂ Y W ) − δ ( α ∂ X W ∂ X Φ + α ∂ Y W ∂ Y Φ)+ (1 + δ W )( − µW − δ µW − ( ∂ X Φ) − ( ∂ Y Φ) − δ − α ∂ X Φ − δ − α ∂ Y Φ) , (23) ∂ T Φ = − βµW − δ βµW − α ( ∂ X Φ) − α ( ∂ Y Φ) + δ − ( ∂ X Φ + ∂ Y Φ)+ δ ( α ∂ X W + α ∂ Y W ) + 2 δ ( ∂ X W ∂ X Φ + ∂ Y W ∂ Y Φ)1 + δ W , (24)where (
X, Y, T ) = ( δx, δy, δ t ). The leading-order solution of the W equation is W = − ( α ∂ X Φ + α ∂ Y Φ)2 δ µ + O (1) . (25)Refining this solution up to O (1) gives W = − ( α ∂ X Φ + ( α + α ) ∂ X ∂ Y Φ + α ∂ Y Φ)4 µ − ( ∂ X Φ) + ( ∂ Y Φ) µ − α ∂ X Φ + α ∂ Y Φ2 δ µ + O ( δ ) . (26)10quation (26) can be used to eliminate W from the equation (24) for Φ. Doing so yields ∂ T Φ = (1 + α β ) ∂ X Φ + (1 + α β ) ∂ Y Φ δ + ( βα − α ) ∂ X Φ + (cid:0) β ( α + α ) − α α (cid:1) ∂ X ∂ Y Φ + ( βα − α ) ∂ Y Φ2 µ + ( β − α )( ∂ X Φ) + ( β − α )( ∂ Y Φ) + O ( δ ) . (27)We have now found the coefficients for the nonlinear terms in Equation (20). The linear coef-ficients of (20) and (27) are equal on the BFN neutral stability curve 1 + α β = 1 + α β = 0 andmatch the form of the isotropic phase equation given in [30].As in (20), the anisotropy allows for the possibility of a Benjamin-Feir-Newell type instabilityto occur in the X or Y directions individually, or for both directions to become unstable simulta-neously. This naturally leads to codimension-one and codimension-two phase equations.The solutions of the phase equation can be compared to the solutions of the phase of solutionsof the ACGLE, as seen in Figure 5 which shows isotropic phase chaos. The phase of A should looksimilar to the solution of the phase equation Φ. We can use the approximation (25) for W (Φ) aftersolving the phase equation for Φ and recreate solutions of the ACGLE by substituting W (Φ) andΦ into the ansatz (21) using (22).Figure 5: A solution A ( x, y, t ) to the ACGLE with parameters α = α = − . β = 5 wassimulated on a square domain [ − , × [ − , t = T /δ ≈ | A | and (b) phase angle angle ( A ) of A . (c) A solution Φ( X, Y, T ) of the phase equation (27)with δ = 0 . T = 50 with the same parameters used to solve A . The Φ calculated fromthe phase equation displays qualitatively the same behavior as the phase of the ACGLE with theseisotropic parameters. (d) The absolute value | A | for A constructed from Φ using Equations (21),(22) and (25). For the codimension-one case, the condition for the y -direction to be BFN-stable is that 1+ α β > α β , we consider the situation in which we are close to the BFN-stability boundary1 + α β = 0. Extending again the scaling used in [12] for the 1D CGLE to our 2D anisotropic casewe unfold this degeneracy by setting 1 + α β = κδ , (28)11here κ is treated as O (1)-parameter. Note that for κ < ) in Figure 2. Since we have BFN stability in the y direction, we have to balance the δ -scalingof t by a δ -scaling of y . Thus, we set ˜ Y = δY = δ y , and use Y = ˜ Y /δ in (27). Omitting the tilde,we arrive at ∂ T Φ = κ∂ X Φ + (1 + α β ) ∂ Y Φ + ( βα − α ) ∂ X Φ2 µ + ( β − α )( ∂ X Φ) + O ( δ ) , (29)with κ as a control parameter. Assuming β (cid:54) = 0, using α = − /β + O ( δ ) in the fourth-orderderivative and nonlinear terms, and omitting O ( δ ) gives the final codimension-one phase equation ∂ T Φ = κ∂ X Φ + (1 + α β ) ∂ Y Φ − µ (cid:18) β (cid:19) ∂ X Φ + ( β + 1 β )( ∂ X Φ) , (30)which retains the nonlinear dispersion parameter β and the y -direction linear dispersion parameter α from the ACGLE (1). We emphasize again that for (30) to be applicable it is required that1 + α β be positive and O (1) so that the Y -diffusion coefficient stays positive and no fourth-orderlinear derivative terms or nonlinear terms with respect to Y are needed to saturate the instability.Assuming κ < T = − κ ˆ µT, ˆ X = (cid:112) ˆ µX, ˆ Y = (cid:115) − κ ˆ µ α β Y, ˆΦ = − κ (cid:0) β + 1 β (cid:1) Φ , where ˆ µ = − κµ /β , all coefficients in (30) become normalized, and the equation simplifies to (withthe hats omitted) ∂ T Φ = − ∂ X Φ − ∂ X Φ + 12 ( ∂ X Φ) + ∂ Y Φ . (31)The fact that all coefficients in the codimension-one phase equation can be normalized means thatthere is a unique phase dynamics (apart from varying initial conditions) that controls the ACGLE-dynamics for generic 1 + α β > α β . Note thatEquation (31) is just the standard 1D KS-equation augmented by a diffusion-term in Y .For constructing approximative solutions of the ACGLE from solutions to the codimension-onephase equation we choose to retain the parameters in Equation (30) so that its solutions may bedirectly compared to solutions of the ACGLE with the same parameters; see Figure 6.The rescaled codimension-one phase equation (31) coincides with the equation derived by Rostand Krug [35] who describe it as an “elastically coupled chain” of one-dimensional Kuramoto-Sivashinsky systems. They derive an equation similar to (31) to analyze the “pinching length” ofthe patterns produced by their version of the aKS equation, which is only anisotropic in the 2nd-order derivative terms and the nonlinear terms. They consider the case for which the 4th-orderspatial derivatives are isotropic. For the codimension-two case, both of the second-order derivative terms in (27) become unstablesimultaneously. We unfold this degeneracy, which occurs when 1 + α β and 1 + α β are bothzero, by setting 1 + α j β = κ j δ for j = 1 ,
2. Using α j = − /β + O ( δ ) in the fourth-order12igure 6: A solution A ( x, y, t ) to the ACGLE with parameters α = ( − δ − /β ≈ − . α =1 /β = 0 . β = 5, δ = 0 .
3, and κ = − − , × [ − , t = 30 /δ ≈ | A | and (b) phase angle(A) . (c) The solution Φ ofthe phase equation (30) using the same parameters at time T = 30. The solution Φ displays similarbehavior as the phase of the ACGLE, but with fewer pinches due to the effect of the nonlinear termand how it affects the pinching length [35]. The computational domain is larger than in Figures 5and 8 because the smaller domain resulted in zero pinches and perfectly vertical ripples. (d) | A | for A approximated from Φ using Equations (21), (22) and (25).derivative terms and the nonlinear terms of (27) and truncating this equation at O (1) then givesthe codimension-two phase equation ∂ T Φ = κ ∂ X Φ + κ ∂ Y Φ − (cid:18) β (cid:19) ∂ X Φ + 2 ∂ X ∂ Y Φ + ∂ Y Φ2 µ + ( β + 1 β ) (cid:0) ( ∂ X Φ) + ( ∂ Y Φ) (cid:1) . (32)This equation is an anisotropic version of the (isotropic) Kuramoto-Sivashinsky (KS) equation (2)with parameters related to the ACGLE. The anisotropy is revealed in the second-order derivativeterms, while the fourth-order derivative terms and the nonlinear terms are still isotropic since thecodimension-two degeneracy occurs with α = α . As a consequence, so called cancellation modesleading to blow-up solutions are not possible for (32), in contrast to the more general anisotropicKS equation studied in [21] and [35], in which the two nonlinear terms may have different signs.To explore the ( κ , κ )-parameter plane numerically, we set ( κ , κ ) = ρ (cos θ, sin θ ). Assuming ρ >
0, the rescaling ˆ T = ρ ˆ µT, ( ˆ X, ˆ Y ) = (cid:112) ˆ µ ( X, Y ) , ˆΦ = 2 ρ (cid:0) β + 1 β (cid:1) Φ , with ˆ µ = µρ /β , simplifies (32) to (with the hats omitted) ∂ T Φ = cos( θ ) ∂ X Φ + sin( θ ) ∂ Y Φ − (cid:0) ∂ X Φ + 2 ∂ X ∂ Y Φ + ∂ Y Φ (cid:1) + 12 (cid:0) ( ∂ X Φ) + ( ∂ Y Φ) (cid:1) . (33)Two examples of numerical solutions of the rescaled codimension two phase equation (32) aredepicted in Figure 7 where both θ values are in the third quadrant so that κ , κ are both negative.The codimension-two phase equation can also be used to recreate solutions of the ACGLE.As in the codimension-one case, we choose to retain the parameters in Equation (32) for thatpurpose so that its solutions may be directly compared to solutions of the ACGLE with the sameparameters. An example of this is shown in Figure 8. The phase of the solution A to the ACGLE13nd the solution Φ to the phase equation display qualitatively the same behavior. Using the lowestorder approximation of W together with Φ yields a recreation of a ACGLE solution A . Using thephase equation gives considerable computational time savings. The ACGLE was simulated to time t ≈ T = 40. Similar times savings appear in the otherphase equations, as seen in Figures 5 and 6. Time savings would be much greater for smaller δ since t = T /δ .Figure 7: Snapshots of Φ from (32) for (a) an isotropic case with θ = 1 . π and (b) an anisotropiccase with θ = 1 . π . Both have β = 3. The angle 1 . π is in the isotropic region with both secondorder derivative terms equally unstable (cos θ = sin θ ≈ − . θ = 1 . π , there is stillinstability in both second order terms, but since (cos θ, sin θ ) ≈ ( − . , − .
6) the instability in the y -direction is much weaker than that in the x -direction leading to the phase chaotic cells becomingelongated along the y -direction. If this direction would be made sufficiently stable, we would seephase-chaotic ripples instead of stretched cells. We have derived and studied Kuruamoto-Sivashinsky (KS)-type phase equations that govern thedynamics of the anisotropic complex Ginzburg-Landau equation near Benjamin-Feir-Newell (BFN)instabilities. While in the isotropic case there is just one type of BFN instability determined by1 + αβ = 0 ( α = α = α ), the anisotropy induces two instability surfaces defined by 1 + α β = 0and 1 + α β = 0, respectively.If only one of these equations is satisfied, an instability (or bifurcation) of codimension one isencountered. In this case the resulting phase equation governing the Ginzburg-Landau dynamicsnear such an instability is the 1D KS equation with respect to one of the two directions, with anadditional diffusion term in the other direction (equation (30) with its rescaled version (31)).If both equations above are satisfied, an instability of codimension two occurs and at the inst-bility the anisotropic Ginzburg-Landau equation degenerates to its isotropic version. In this case14igure 8: A solution A ( x, y, t ) to the ACGLE with parameters β = 2 and θ = 1 . π , ρ = 1, δ = 0 . α = ( δ ρ cos θ − /β ≈ − . α = ( δ ρ sin θ − /β ≈ − . − , × [ − , t = 40 /δ ≈ | A | and (b) phase angle(A) . (c) The solution Φ of the phase equation (32) using the same parametersat time T = 40. The solution Φ displays similar behavior as the phase of the ACGLE, albeit notexact due to the size of δ . (d) | A | for A approximated from Φ using Equations (21), (22) and (25).The solution to Φ exhibits fewer pinches than the solution to A at the given times, however thesnapshots in (b,c) become closer to (a,b) if run to longer time. Similarly, earlier times of (a,b) moreclosely resemble (b,c). The effective strength of the nonlinearity, dependent on β , and which causesthe pinches appear, is not the same for the given parameters.the resulting phase equation is the anisotropic, two-dimensional KS equation (32) with its rescaledversion (33), whose anisotropy is revealed in the second order derivative terms and quantified bythe angle θ while the other terms are isotropic.Attempts to generate solutions of the Ginzburg-Landau equation from solutions of these phaseequations were successful in both cases. A paper on a systematic parameter study of the solutionsof the anisotropic complex Ginzburg-Landau equation is in preparation, including parameters awayfrom the BNF-instability surfaces.From a general pattern formation point of view, the anisotropic complex Ginzburg-Landauequation is the generic amplitude equation for oscillatory (Hopf) instabilities with zero wavenum-bers in anisotropic extended systems with reflection symmetries in both directions, i.e. the basicstate of the system becomes neutrally stable with respect to bulk oscillations. There are threetypes of instabilities with nonzero wavenumbers (one stationary and two oscillatory; see [10]) forwhich a system of two or four coupled Ginzburg-Landau equations becomes the generic system ofamplitude equations. For these coupled Ginzburg-Landau equations, BNF-type instabilities as wellas Eckhaus-type instabilities result in coupled phase equations. Specifically, if the instability of thebasic state is oscillatory (Hopf-type), the Ginzburg-Landau system contains global coupling terms[10] leading to global coupling terms in the resulting coupled phase equations. These coupled phaseequations are the subject of current studies.All simulations were computed with the authors’ own codes based on [22], which can be foundat https://github.com/drhandwerk/ACGLE-Phase-Equations.15 cknowledgements This work was supported at Colorado State University by NSF grant DMS-1615909 to I. Oprea,G. Dangelmayr, and P. D. Shipman.
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