Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves
TTURING PATTERNS IN PARABOLIC SYSTEMS OF CONSERVATION LAWSAND NUMERICALLY OBSERVED STABILITY OF PERIODIC WAVES
BLAKE BARKER, SOYEUN JUNG, AND KEVIN ZUMBRUN
Abstract.
Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservationlaws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use theseconditions to find families of periodic solutions bifurcating from uniform states, numerically contin-uing these families into the large-amplitude regime. For the examples studied, numerical stabilityanalysis suggests that stable periodic waves can emerge either from supercritical Turing bifurca-tions or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations.This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of con-servation laws can occur. Determination of a full small-amplitude stability diagram– specifically,determination of rigorous Eckhaus-type stability conditions– remains an interesting open problem. Introduction
The study of periodic solutions of conservation laws and their stability, initiated in [OZ03a,OZ03b] and continued in [Ser05, JZ10], etc., has led to a number of interesting developments,particularly in the related study of roll-waves in inclined shallow-water flow. For an account ofthese developments, see, e.g., [JNRZ12] and references therein. However, in the original context ofconservation laws, so far no example of a stable periodic wave has been found.
Indeed, one of theprimary results of [OZ03a, PSZ13] was that for the fundamental example of planar viscoelasticity,stable periodic waves do not exist, due to a special variational structure of this particular system;it was cited as a basic open problem whether stable periodic waves could arise for any system ofconservation laws, either physically motivated: or artificially contrived.In the more standard context of reaction diffusion systems and classical pattern formation the-ory, by contrast, stable periodic solutions are abundant and well-understood, through the mech-anism of
Turing instability , or bifurcation of small-amplitude, approximately-constant period, pe-riodic solutions from a uniform state. For such waves, stability is completely determined by anassociated
Eckhaus stability diagram , as derived formally in [Eck65] and verified rigorously in[Mie95, Mie97, Sch96, SZJV16], essentially by perturbation from constant-coefficient linearizedbehavior. By contrast, the small-amplitude waves investigated up to now (see Remark 3.2) comethrough more complicated zero-wave number bifurcations in which period goes to infinity as am-plitude goes to zero and the stability analysis is far from constant-coefficient (see, e.g., [Bar14] inthe successfully-analyzed case of shallow-water flow).Our simple goal in this paper, therefore, is to seek stable periodic waves via a conservation lawanalog of Turing instability.
In the first part, we find an analog of Turing instability, with whichwe are able to generate large numbers of examples of spatially periodic solutions of conservationlaws. Next, we find an interesting dimensional restriction to systems of three or more coordinates,
Date : July 12, 2018.Research of B.B. was partially supported under NSF grant no. DMS-1400872.Research of S.J. was partially supported by the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MSIP) (No. 2016009978).Research of K.Z. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745. a r X i v : . [ m a t h . A P ] J a n xplaining the absence of Turing instabilities for 2 × × Turing instability for conservation laws
We begin by defining a notion of Turing instability for systems of conservation laws(2.1) u t + f ( u ; ε ) x = ( D ε u x ) x ,u ∈ R n , where ε is a bifurcation parameter and D for simplicity is taken constant. Linearizing (2.1)about a uniform state u ( x, t ) ≡ u yields the family of constant-coefficient equations(2.2) u t = L ( ε ) u := − A ε u x + D ε u xx with dispersion relations λ j ( ξ ) ∈ σ ( − iξA ε − ξ D ε ), ξ ∈ R , where σ ( · ) here and elsewhere denotesspectrum of a matrix or linear operator. The state u is spectrally (hence nonlinearly) stable if(2.3) (cid:60) σ ( − iξA ε − ξ D ε ) ≤ − θ | ξ | , θ > , for all ξ ∈ R [Kaw83].Following the original philosophy applied by Turing [Tur52] to reaction diffusion systems, weseek a natural set of conditions guaranteeing low- and high-frequency stability – i.e., that (2.3) holdfor | ξ | → , ∞ – but allowing instability at finite frequencies | ξ | (cid:54) = 0 , ∞ . Should this be possible,then performing a homotopy in ε between stable and unstable states, we may expect generically toarrive at a special bifurcation point ε = ε ∗ , without loss of generality ε ∗ = 0, for which (2.3) holdsuniformly away from special points ξ = ± ξ ∗ , at which(2.4) max ξ (cid:54) =0 (cid:60) σ ( − iξA ε ∗ − ξ D ε ∗ ) = 0is achieved (note, by complex conjugate symmetry, that extrema appear in ± pairs) and for which(2.3) fails strictly as ε is further increased. We may then conclude, by standard bifurcation theoryapplied to the domain of periodic functions with period X := 2 π/ξ ∗ the appearance of nontrivialspatially periodic solutions with periods near X , similarly as in the reaction diffusion case [Mie95,Mie97, Sch96, SZJV16].At ξ = 0, (2.3) yields that A is hyperbolic, in the sense that it has real semisimple eigenvalues.Without loss of generality, therefore, take A diagonal, with entries a j , j = 1 , . . . , n . In the sim-plest case that A is strictly hyperbolic, in the sense that these a j are distinct, we find by spectralperturbation expansion about ξ = 0 [Kaw83] that the corresponding eigenvalue expansions are λ j ( ξ ) = − ia j ξ − D jj ξ + O ( ξ ) , o that (2.3) ( ξ (cid:28)
1) is equivalent to the condition that D have positive diagonal entries D jj .Similarly, by spectral expansion about ξ = ∞ , σ ( − iξA − ξ D ) = − ξ σ ( D ) + O ( ξ ) , so that (2.3)( ξ = ∞ ) is equivalent to the condition that D be unstable, i.e., have eigenvalues withstrictly positive real part . Collecting, our conditions are ( C ) : • A ε is diagonal with distinct entries, and • D ε has positive diagonal entries and eigenvalues with strictly positive real part.These are to be contrasted with Turing’s conditions in the reaction diffusion case u t = Du xx + g ( u )that D be symmetric positive and A := dg ( u ) be symmetric negative definite [Tur52].2.1. Turing instability and Hopf bifurcation.
Let (2.4) hold, with λ = ± iτ ∈ σ ( − iξA − ξ D )for ξ = ± ξ ∗ , ξ ∗ (cid:54) = 0. Then, changing to the moving coordinate frame x → ˜ x := x − ct , for c := τ /ξ ∗ , or, equivalently, under the the change of coordinates A → ˜ A := A − cI , we have λ = 0 ∈ σ ( − iξ ˜ A − ξ D ) for ξ = ± ξ ∗ , i.e., det( − iξ ˜ A − ξ D ) = 0 at ξ = ξ ∗ , or(2.5) ± iξ ∗ ∈ σ ( D − ˜ A ) . Condition (2.5) may be recognized as the condition for Hopf bifurcation of an equilibrium u ( x, t ) ≡ constant of the traveling-wave ODE(2.6) D ε u (cid:48) = f ( u ; ε ) − cu + q, where q is a constant of integration, for which the linearized equation is u (cid:48) = D − ˜ Au, ˜ A again diago-nal. Thus, we recover by finite-dimensional bifurcation theory the previously-remarked appearanceof nontrivial periodic solutions with period near X = 2 π/ξ ∗ . We also obtain the alternative bifur-cation criterion (2.5). This simplifies the problem a great deal; for one thing, we are now workingwith real matrices, as occur for symbols in the reaction diffusion case, and not complex ones.2.1.1.
Dimensional count.
From the usual Hopf bifurcation theorem for ODE, we find that for eachfixed nearby q , c , there exists a one-parameter family of nontrivial periodic solutions bifurcatingfrom the constant solution, generically parametrized nonsingularly by period X . Thus, fixing q = 0,we obtain a 2 -parameter family of periodic solutions, generically well-parametrized by c and X . Finding Turing instabilities.
To find Turing instability, we may seek A ε and D ε satisfying( C ) , ε ∈ R a bifurcation parameter, such that (2.4) is violated at ε = 0 (instability), but (2.3) issatisfied for all ξ at ε = 1 (stability), for example if A = Id or D = Id. For, in this case, theconditions ( C ) on A ε , D ε insure that at the largest value ε ∗ of ε for which (2.4) is satisfied, themaximum (2.4) is achieved at some ξ = ξ ∗ (cid:54) = 0, while for ε > D = I , in which case the spectra of ( − iξA − ξ D )are simply λ j ( ξ ) = − iξa j − ξ ; nor can (2.5), since σ ( ˜ A ) is by assumption real. Thus, we suggest,first, finding examples ˇ A , ˇ D satisfying (2.5) either analytically or by checking random matrices,then, setting up a homotopy D (cid:15) := (cid:15) ˇ D + (1 − (cid:15) ) I from the identity to ˇ D . Since, as just observed, σ ( − iξ ˇ A − ξ D (cid:15) ) is stable for ε = 0, while for ε = 1 it is at most neutrally stable, having zeroeigenvalues at ξ = ± ξ ∗ (cid:54) = 0, we find that for some (cid:15) ∈ (0 , σ ( − iξ ˇ A − ξ D (cid:15) ) is exactly neutral,i.e., a Turing instability , with eigenvalues ± iτ at ξ = ± ˆ ξ ∗ (note: different from the original ξ ∗ ingeneral!). As described above, this corresponds to a Hopf bifurcation in the traveling-wave ODEfor speed c ∗ := τ / ˆ ξ ∗ , with limiting wave number ˆ ξ ∗ and period X ∗ := 2 π/ ˆ ξ ∗ . . Negative results
We next describe situations in which Turing instability cannot occur, narrowing our search.3.1.
The × case. We have the following result for n = 2, strikingly different from the situationof the reaction diffusion case. Proposition 3.1.
Assuming ( C ) , there exist no Turing-type instabilities of (2.1) for n = 2 .Proof. Take by assumption A diagonal. Since D − A is real, appearance of a pure imaginary eigen-value iτ implies the appearance also of its complex conjugate − iτ , hence trace is zero and deter-minant is positive. By a scaling transformation S = (cid:18) α β (cid:19) not affecting diagonal form of A , wemay arrange therefore that D − A = (cid:18) c − − c (cid:19) =: J, for some c <
1. Noting that J = ( c − I, we may solve to obtain D = c − AJ = c − (cid:18) a c a − a − a c (cid:19) . The requirement that D have positivediagonal implies, with c <
1, that a c < a c >
0, so that a and a have opposite sign. But,det D = ( c − − a a (1 − c ) > a and a have the same sign, hence these twoconditions cannot hold at once. (cid:3) Example 3.2.
The viscoelasticity model τ t − u x = d τ xx , u t + p ( τ ) x = d u xx studied by Oh-Zumbrun [OZ03a] falls into the above framework, hence does not admit Turing instabilities. In fact,periodic waves arise in this model through Bogdanov-Takens bifurcation associated with splittingof two or more equilibria, a more complicated bifurcation far from constant-coefficient behavior.3.2. Simultaneous symmetrizability.
Another case in which Turing instabilities do not occuris when A and D are simultaneously symmetrizable, or, equivalently, can be converted by change ofcoordinates to be both symmetric. For, then, in the new coordinates, D , being symmetric positivedefinite, has a square root, and so D − A is similar to the symmetric matrix D / D − AD − / = D − / AD − / , hence has real eigenvalues. More generally, it is easy to see that Turing instabilitydoes not occur for A symmetric and (cid:60) D := (1 / D + D T ) >
0, since D − Av = iτ v would imply0 = (cid:60) iτ (cid:104) v, Av (cid:105) = Re (cid:104) v, Dv (cid:105) = (cid:104) v, (cid:60) Dv (cid:105) > , a contradiction. This recovers the well-known factthat existence of a viscosity-compatible convex entropy for the system (2.1) implies nonexistenceof non-constant stationary solutions, since existence of such an entropy implies the correspondingsymmetry conditions on the linearized equations. Thus, taking A without loss of generality diagonal,we must specifically seek D nonsymmetric , D + D T nonpositive in order to find Turing instability.3.3. Nonstrict hyperbolicity.
Finally, we give a simple example showing that the condition ofstrict hyperbolicity of A ε is necessary in ( C ) . Consider the matrices(3.1) A ε = ε
00 0 1 and D = − . Here, σ ( D ) = { } ; so − iξA − ξ D is stable for | ξ | → + ∞ . For | ξ | →
0, we look at 2 × A and D ,(3.2) ˜ A = (cid:18) (cid:19) and ˜ D = (cid:18) (cid:19) . Then, the two eigenvalues of − iξA − ξ D close to iξ for ξ (cid:28) l j ( ξ ) = − iξ − ξ ˜ d j , where ˜ d j are eigenvalues of ˜ D . We easily see that ˜ D has two realeigenvalues with opposite sign because det ( ˜ D ) = − <
0. Thus, (2.3) is not satisfied for | ξ | → emark . Though example (3.1), failing ( C ) , does not itself yield Turing instability, it is quiteuseful in finding nearby systems that do. For, note perturbation in ε generates matrices D − A with nonstable eigenvalues despite A >
0. Perturbing first ε to obtain instability, then A still moreslightly to recover strict hyperbolicity, we thus obtain an example satisfying ( C ) with unstable D − A , which yields a Turing bifurcation upon homotopy D → I . We in fact used this method togenerate the examples of Section 5. (We have generated other examples in other ways, that werenot reported here; all exhibited similar behavior, however.)4. Spectral and nonlinear stability
Before describing our numerical investigations, we briefly recall the abstract stability frameworkdeveloped in [OZ03a, JZ10, JNRZ12], etc., relevant to stability of the nontrivial periodic wavesbifurcating from a constant solution at Turing instability. First, recall [JZ10, JNRZ12] that, underthe condition of transversality of the associated periodic orbit of the traveling-wave ODE (guaran-teed in this case by the Hopf bifurcation scenario, for sufficiently small-amplitude waves), nonlinearstability with respect to localized perturbations of the periodic wave considered as a solution on thewhole line is determined (up to mild nondegeneracy conditions) by conditions of diffusive spectralstability, as we now describe.By Floquet theory, the L ( R ) spectrum of the linearized operator L about a periodic waveof period X is entirely essential spectrum, corresponding to values λ ∈ C for which there existgeneralized eigenfunction solutions v ( x ) = e iξx w ( x ), ξ ∈ R , of the associated eigenvalue equation( L − λ ) v = 0 with w periodic, period X . The dissipative stability conditions are that this spectrumhave real part ≤ − ηξ , η >
0, for all ξ ∈ R , and strictly negative for ( ξ, λ ) (cid:54) = (0 , ε bounded away from ε ∗ , the spectra near ( ξ, λ ) = (0 ,
0) consists ofthe union of ( n + 1) smooth spectral curves λ j ( ξ ) = − ia j ξ + o ( ξ ) through the origin λ = 0, which,under the nondegeneracy condition that a j be distinct, are analytic in ξ , admitting second-orderexpansions(4.1) λ j ( ξ ) = − ia j ξ − b j ξ + O ( ξ ) , j = 1 , . . . , n + 1 . Moreover, the functions λ j ( ξ ) correspond to the linearized dispersion relations for the associatedsecond-order Whitham system , an associated second-order ( n + 1) × ( n + 1) system of conservationlaws formally governing slow modulational behavior [Whi11, Ser05, JNRZ12]. Thus, low-frequencydiffusive spectral stability is equivalent to well-posedness (hyperbolic-parabolicity) of the Whithamsystem, which is in turn equivalent to reality of a j (hyperbolicity) and positivity of (cid:60) b j (parabol-icity) in (4.1), with high-frequency spectral stability given by (cid:60) λ ≤ − η < | ξ | ≥ η , η > X ∗ such that the wave-numbers ± ξ ∗ at ε = ε ∗ are equal to zero modulo 2 π/X ∗ , we find by direct Fourier transform calculation that theconstant solution at ε = ε ∗ has low-frequency spectrum consisting of ( n + 2) spectral curves passingthrough the origin, with all other spectra satisfying (cid:60) λ ≤ − η < η >
0. The spectra ofthe bifurcating periodic waves perturbs smoothly from these values as ε is increased, hence high-frequency diffusive stability is guaranteed. However, low-frequency stability is now determinedby a possibly complicated bifurcation of ( n + 2) spectral curves involving the ( n + 1) “Whithamcurves” (4.1) passing through the origin plus an additional curve originating from the constantlimit passing close to but not through the origin. These curves are clearly visible in the numericallyapproximated spectra displayed below in Section 5 for example systems with n = 3: namely, 4Whitham curves passing through the origin, with a 5th (initially) neutral spectral curve passingnear the origin, with all 5 of these passing through the origin at the bifurcation point ε = ε ∗ . . Numerical investigations
Guided by the results of Sections 2, 3, and 4, we now perform the main work of the paper,carrying out numerical existence and stability investigations for periodic solutions of systems ofconservation laws arising through Turing bifurcation from the uniform state in dimension n = 3.Numerics are carried out using the MATLAB-based package STABLAB developed for this purpose[BHLZ].5.1. Quadratic nonlinearity.
We first consider the system(5.1) u t + A ε u x + N ( u ) x = Du xx , with(5.2) A ε := a + ε
00 0 3 , D := − , and N ( u ) := β u , where a = 2 . ε is a bifurcation parameter that we will vary and u ≡ u = 0, we have(5.3) u t + A ε u x = Du xx . We first check Turing-type instability conditions for u ≡ A ε is strictlyhyperbolic and D has positive diagonal entries with σ ( D ) = { } , which means that − iξA ε − ξ D is stable near ξ = 0 or ξ = ±∞ . We examine numerically stability of u ≡ ε changes. In Figure1, we plot the spectrum of − iξA ε − ξ D with ε = − . ε = 0, and ε = 0 .
2. It is seen that theconstant solution u ≡ ε < ε >
0. Thus, Turing instability occursat ε = 0, that is, (2.4) is satisfied with ± iτ ∈ σ ( − iξA − ξ D ) for τ ≈ . ξ ∗ ≈ ± .
16. As weobserved in the previous section, ± iξ ∗ are eigenvalues of D − ( A − c ∗ I ) for c ∗ = τξ ∗ ≈ .
30. So thecondition for Hopf bifurcation of a constant solution u ≡ − cu + A ε u + N ( u ) = Du (cid:48) + q is satisfied at the bifurcating point ε = 0 and c = c ∗ . Here q ∈ R is an integration constant andwe fix q = 0 from now on. In Figure 2, we plot the spectrum of − iξ ( A ε − c ∗ I ) − ξ D for the same ε as in Figure 1, showing how this moves the neutral spectrum from λ = ± iτ to λ = 0.( a ) ( b ) ( c ) Figure 1.
Plot with dots of a sampling of the spectrum of the constant solution, − iξA − ξ D , with (a) ε = − .
2, (b) ε = 0, (c) ε = 0 .
2. The dashed vertical linemarks the imaginary axis.The Hopf bifurcation leads to periodic profiles bifurcating from the uniform state u ≡
0. Inorder to solve for these profiles, we let ε be a free variable and vary the period X and wave speed c ,approximating associated solutions using the periodic profile solver built into STABLAB, which usesMATLAB’s Newton-based boundary-value problem solver bvp5c. In addition to periodic boundary a ) ( b ) ( c ) Figure 2.
Plot with dots of a sampling of the spectrum of the constant solution, − iξ ( A − c ∗ I ) − ξ D , with (a) ε = − .
2, (b) ε = 0, (c) ε = 0 . c = c ∗ ≈ . w · f ( y (0)) = 0 where y (cid:48) ( x ) = f ( y ( x )) is theprofile ODE ((2.6) in the present case) and w is a random vector. Unless w is a degenerate choice, w · ˙ y ( t ) = 0 for some t by periodicity of y and Rolle’s Theorem, so this phase condition choosesa solution (at least locally) uniquely. To numerically solve the profile equation with a quadraticnonlinearity, we first obtain a solution by using as an initial guess u ( x ) = √ ε (cid:60) ( e πix v ) /
10, where v is the real part of an eigenvector, whose corresponding eigenvalue has non-zero imaginary part, ofthe profile Jacobian evaluated at the fixed point (0 , , T . That is, we start with an initial guessconsisting of a strategically scaled periodic solution of the linearized equations at the bifurcationpoint ε = 0. Once we have a profile solution via this guess, we use continuation to solve forother profiles with nearby period X and speed c , obtaining thereby a full 2-parameter family ofapproximate solutions parametrized by ( c, X ), as described in Section 2.1.1.In Figure 3 (a) and (b), we plot the stability bifurcation diagram in the coordinates of shiftedwave speed c = c − c ∗ and period X . The bifurcation diagram shows that there is a family of stablewaves bifurcating from the Turing bifurcation. There is a small region of instability occurring froma “parabolic” Whitham instability, or change in curvature of a neutral spectral curve through theorigin, corresponding to negative diffusion or ill-posedness of the associated formal slow modulationWhitham equations, which separates the region of stability near the Turing bifurcation point andthe larger stability region. Figures 3 (d)-(f) demonstrate this onset of Whitham-type instability asseen in the spectrum of the bifurcating periodic waves. In Figure 3 (c), we see that the spectrumof the background constant solution becomes unstable as ε increases, so that the periodic profileshown in Figure 3 (g) comes into existence through a super-critical Hopf bifurcation. Finally, inFigure 3 (g), we plot the periodic profile for β = − ε = 2 . e − c = c ∗ + 4 . e − X = 5 . ε nearby ε ∗ . It explains why the the spectrum of stable periodic wavesbifurcating from Turing bifurcation in Figure 3 (d) has an additional 5th curve which is very close tothe origin but not through the origin. Stability of small-amplitude waves is determined by behaviorof these 5 neutral curves, either by movement of the maximum real part of the 5th curve into theunstable or stable half-plane (“co-periodic” stability, corresponding with super- or sub-criticality ofthe associated Hopf bifurcation), or by a “Whitham-type” instability consisting of loss of tangency a ) ( b ) ( c )( d ) ( e ) ( f )( g ) x -4-2024 y ( x ) × -3 y y y Figure 3. (a) Stability bifurcation diagram in the coordinates of shifted wave speed c = c − c ∗ and period X . Pink dots (light dots in grayscale) and black dotscorrespond respectively to stable and unstable waves. (b) Zoom in of (a) showinga family of stable waves in parameter space leading to the point of the Turingbifurcation. There is a small region of instability separating the stable waves nearthe Turing bifurcation point and the large stability region. (c) Plot of the spectrumof the zero constant solution when ε = 2 . e − c = c ∗ + 4 . e −
3, and X =5 .
44, indicating that the Turing bifurcation corresponds to a supercritical Hopfbifurcation. (d) Plot of the spectrum of a periodic wave in the family of stablewaves bifurcating from the Turing bifurcation. (e) Plot of the spectrum of a periodicwave in the family of unstable waves separating the two regions of stability. (f) Plotof the spectrum of a periodic wave in the large stability region. (g) Plot of thebifurcating periodic profile when ε = 2 . e − c = c ∗ + 4 . e −
3, and X = 5 . β = −
10 and a dashedline marks the imaginary axis.to the imaginary axis (first-order, or “hyperbolic” instability) or change in curvature (2nd order,or “parabolic” instability) of one of the 4 neutral curves through the origin; see Section 4.For the quadratic nonlinearity, if u ( x ) is a profile solution for a fixed β , then − u ( x ) is a profilesolution for − β , with the same value of ε . Thus, we are not able to produce a correspondingsub-critical Hopf bifurcation by reversing the sign of β , but a mirror super-critical bifurcation.To find examples of stable periodic profiles corresponding to both sub and super-critical Hopfbifurcations, we change the quadratic nonlinearity to a cubic nonlinearity in the next example,removing this symmetry and allowing us to change from super- to sub- by changing the sign of β . .2. Cubic nonlinearity.
We consider next the system of conservation laws(5.5) u t + A ε u x + N ( u ) x = Du xx , with(5.6) A ε := a + ε
00 0 3 , D := − , and N ( u ) := β u , where a = 2 . ε as a bifurcationparameter. The stability of u ≡ ε varies is already shown in Figure 1 and Figure 2.Starting from the super-critical periodic profile solutions found previously for the quadraticnonlinearity, we obtain a solution for the cubic nonlinearity by continuation in a homotopy variable0 ≤ h ≤ N ( U ) = [ β ( hy + (1 − h ) y ) , , T . To obtain a sub-critical profilesolution for the cubic nonlinearity, we use the approximate symmetry ( β, c, ε ) → ( − β, − c, − ε ),which is valid at the linear periodic level only. Thereafter, we solve for profiles using continuation.In Figure 4, we plot the bifurcating stable periodic solution through a super-critical Hopf bifur-cation. Since ε > β = 10, c = 0 . X = 6 , and ε = 8 . e −
1. InFigure 4 (a), we plot a stability diagram in the coordinates of shifted wave speed c = c − c ∗ andperiod X . We do not find a family of stable waves bifurcating from the Turing instability.By changing the sign of β , we find the stable periodic solutions through a sub-critical Hopfbifurcation as demonstrated in Figure 5. Since ε < β = − c = − . X = 4 . ε = − . e −
3. In Figure 5 (a), we plot a stability diagram inthe coordinates of shifted wave speed c = c − c ∗ and period X . We do not find a family of stablewaves bifurcating from the Turing instability.( a ) ( b ) -0.4 -0.2 0 0.2 Re( λ ) -1-0.500.51 I m ( λ ) ( c ) x -0.2-0.100.10.20.30.4 y ( x ) y y y Figure 4. (a) Stability diagram in the coordinates of shifted wave speed c = c − c ∗ and period X for β = 10. Pink dots (light dots in grayscale) and black dotscorrespond respectively to stable and unstable waves. (b) For a stable wave, we plotin (b) its spectrum and in (c) the wave itself, with β = 10, c = 0 . X = 6 , and ε = 8 . e −
1. A dashed line marks the imaginary axis in (b).5.3.
Numerical stability method.
To determine the spectrum of the periodic profiles, we usedHill’s method. The associated eigenvalue problem is given by Lv = λv where the linear oper-ator L takes the form L j,k = (cid:80) m jk q =1 f j,k,q ( x ) ∂ q ∂x q . The coefficients f j,k,q ( x ) are X periodic. Asin [DKCK07], we use a Fourier series to represent the coefficient functions f j,k,q , f j,k,q ( x ) = (cid:80) ∞ j = −∞ ˆ φ j,k,q e i πjx/X , and write the generalized eigenfunctions as v ( x ) = e iξx (cid:80) ∞ j = −∞ ˆ v j e iπjx/X , a ) ( b ) -0.4 -0.2 0 0.2 Re( λ ) -1-0.500.51 I m ( λ ) ( c ) x -0.3-0.2-0.100.10.20.3 y ( x ) y y y ( d ) -0.2 -0.15 -0.1 -0.05 c ǫ X = 5.4
Figure 5. (a) Stability diagram in the coordinates of shifted wave speed c = c − c ∗ and period X for β = −
10. Pink dots (light dots in grayscale) and black dotscorrespond respectively to stable and unstable waves. For a stable wave, we plotin (b) its spectrum and in (c) the wave itself, with β = − c = − . X = 4 . ε = − . e −
3. A dashed line marks the imaginary axis in (b). In (d) we plota curve showing existence, up to numerical approximation, of periodic profiles ofperiod X = 5 . c and ε when β = −
10 and the nonlinearity iscubic. A thin horizontal line marks the axis.where ξ ∈ ( − π/ X, π/ X ] is the Floquet exponent. Substituting these quantities into the eigen-value problem and equating coefficients gives an infinite dimensional eigenvalue problem for eachfixed ξ . By truncating the Fourier series at N terms and using MatLabs FFT function to determinethe coefficients ˆ φ j,k,q , we arrive at a finite dimensional eigenvalue problem L ξN ˆ v = λ ˆ v , which wesolve with MATLAB’s eigenvalue solver. All computations were done using STABLAB [BHLZ]. Forfurther information about Hill’s method and its convergence properties, see [CD10, DK06, JZ12].5.4. Computational statistics.
All computations were carried out on a Macbook pro quad coreor a Leopard WS desktop with 10 cores. Computing a profile took approximately 2 seconds or less,and computing the spectrum via Hill’s method took on average 20-60 seconds depending on thenumber of modes used. We typically used 101 Floquet parameters and 41 or 81 Fourier modes whenusing Hill’s method. Each stability diagram took less then 24 hours to compute on the LeopardWS desktop. 6.
Discussion and open problems
We have identified an analog of Turing instability occurring for n × n systems of conservation lawsof dimension n ≥
3, leading to a large family of spatially periodic traveling waves. Our numericalstability investigations give convincing numerical evidence that at least some of these waves are table, answering the question posed in [OZ03a, PSZ13] whether there can exist stable periodicsolutions of conservation laws.Moreover, the same numerical investigations indicate that at least for some model parameters,the bifurcation diagram near Turing instability/Hopf bifurcation includes an open region of insta-bility. This opens the possibility for rigorous proof of existence of stable periodic waves througha small-amplitude bifurcation analysis as carried out in [Mie95, Mie97, Sch96, SZJV16] for thereaction diffusion case. Such an analysis we consider an extremely interesting open problem. Note,however, that it is inherently more complicated than the reaction diffusion version, involving n + 2bifurcation parameters ( X, c, q ), X, c ∈ R , q ∈ R n rather than the two parameters of the reac-tion diffusion case. For an example of intermediate complexity, we point to the recent analyses[MC00, Suk16] of reaction diffusion equations with a single conserved quantity, featuring a three-parameter bifurcation. References [Bar14] Blake Barker. Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin filmflow.
Journal of Differential Equations , 257(8):2950–2983, Oct 2014.[BHLZ] Blake Barker, Jeffrey Humpherys, Joshua Lytle, and Kevin Zumbrun. STABLAB: A MATLAB-basednumerical library for evans function computation. https://github.com/nonlinear-waves/stablab.git.[CD10] Christopher W. Curtis and Bernard Deconinck. On the convergence of Hill’s method.
Math. Comp. ,79(269):169–187, Jan 2010.[DK06] Bernard Deconinck and J. Nathan Kutz. Computing spectra of linear operators using the Floquet–Fourier–Hill method.
Journal of Computational Physics , 219(1):296–321, Nov 2006.[DKCK07] Bernard Deconinck, Firat Kiyak, John D. Carter, and J. Nathan Kutz. SpectrUW: A laboratory for thenumerical exploration of spectra of linear operators.
Mathematics and Computers in Simulation , 74(4-5):370–378, Mar 2007.[Eck65] W. Eckhaus. Studies in nonlinear stability theory.
Springer tracts in Nat. Phil. Vol. 6 , 1965.[JNRZ12] Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun. Nonlocalized modulationof periodic reaction diffusion waves: Nonlinear stability.
Archive for Rational Mechanics and Analysis ,207(2):693–715, Oct 2012.[JZ10] Mathew A. Johnson and Kevin Zumbrun. Nonlinear stability of periodic traveling waves of viscous con-servation laws in the generic case.
Journal of Differential Equations , 249(5):1213–1240, 2010.[JZ12] Mathew A. Johnson and Kevin Zumbrun. Convergence of Hill’s method for nonselfadjoint operators.
SIAM J. Numer. Anal. , 50(1):64–78, Jan 2012.[Kaw83] Shuichi Kawashima.
Systems of a hyperbolic-parabolic composite type, with applications to the equationsof magnetohydrodynamics . PhD thesis, Kyoto University, 1983.[MC00] P C Matthews and S M Cox. Pattern formation with a conservation law.
Nonlinearity , 13(4):1293–1320,2000.[Mie95] A. Mielke. A new approach to sideband-instabilities using the principle of reduced instability. In A. Doel-man and A. van Harten, editors,
Nonlinear Dynamics and Pattern formation in the Natural Environment.Pitman Research Notes in Math , pages 206–222. UK: Longman, 1995.[Mie97] Alexander Mielke. Instability and stability of rolls in the Swift-Hohenberg equation.
Communications inMathematical Physics , 189(3):829–853, Nov 1997.[OZ03a] Myunghyun Oh and Kevin Zumbrun. Stability of periodic solutions of conservation laws with viscosity:Analysis of the Evans function.
Arch. Ration. Mech. Anal. , 166(2):99–166, 2003.[OZ03b] Myunghyun Oh and Kevin Zumbrun. Stability of periodic solutions of conservation laws with viscosity:pointwise bounds on the Green function.
Arch. Ration. Mech. Anal. , 166(2):167–196, 2003.[PSZ13] Alin Pogan, Arnd Scheel, and Kevin Zumbrun. Quasi-gradient systems, modulational dichotomies, andstability of spatially periodic patterns.
Differential Integral Equations . , 26(3/4):389–438, 2013.[Sch96] G. Schneider. Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation.
Comm.Math. Phys. , 178(3):679–702, 1996.[Ser05] Denis Serre. Spectral stability of periodic solutions of viscous conservation laws: Large wavelength anal-ysis.
Communications in Partial Differential Equations , 30(1-2):259–282, Apr 2005.[Suk16] Alim Sukhtayev. Diffusive stability of spatially periodic patterns with a conservation law. arXiv:1610.05395 (preprint) , 2016. SZJV16] Alim Sukhtayev, Kevin Zumbrun, Soyeun Jung, and Raghavendra Venkatraman. Diffusive stability ofspatially periodic solutions of the Brusselator model. arXiv:1608.08476 (preprint) , 2016.[Tur52] A. M. Turing. The chemical basis of morphogenesis.
Philosophical Transactions of the Royal Society B:Biological Sciences , 237(641):37–72, Aug 1952.[Whi11] Gerald Beresford Whitham.
Linear and nonlinear waves , volume 42. John Wiley & Sons, 2011.
Brigham Young University, Provo, UT 84602
E-mail address : [email protected] Kongju National University, Korea
E-mail address : [email protected] Indiana University, Bloomington, IN 47405
E-mail address : [email protected]@indiana.edu