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On the band-width of stable nonlinear stripe patterns in finite size systemsMirko Ruppert1 and Walter Zimmermann1, a)

Theoretische Physik, Universitat Bayreuth, 95440 Bayreuth, Germany

(Dated: 10 November 2021)

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geologicalscales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e.,they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- andzigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size.This enlargement comes about because suppressing degrees of freedom from the two instabilities goes alongwith the system reduction, and the enlargement depends on the boundary conditions, as we show analyticallyand numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segelequation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basicstate is simultaneously stable in certain parameter ranges, which is especially important for Turing pattern incells. In addition, we explain why below a certain system width stripe pattern behave quasi-one-dimensionalin two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stablepatterns for different combinations of boundary conditions.

Nonlinear stripe patterns are ubiquitous in na-ture, and their driving mechanisms are as di-verse as the systems themselves in which theyoccur1–15. Stripe patterns have the universalproperty of being stable at different values of thewavenumber, and these stable wavenumber re-gions are the so-called Busse balloons after theirpioneer1,3,11,16,17. To the instabilities boundingthe stable wavenumber range of stripe patternscount the generic Eckhaus instability, a long-wavelength longitudinal (compressional) instabil-ity, and the zigzag instability, a long-wavelengthtransverse instability1,18. Stable wavenumberranges are restricted even in large systems bypattern suppressing boundary conditions at thedomain sides19,20 or are even selected by spatialinhomogeneities, e.g. via so-called ramps21–23. Incontrast, in short systems the stable wavenum-ber range can be enlarged, as in Ref.24 for quasi-one-dimensional systems predicted and experi-mentally confirmed in Ref.25,26. Such a rangeextension depends on the boundary conditionsand the second spatial dimension as we explainin this work analytically and numerically by in-vestigating the generic Swift-Hohenberg modeland the universal Newell-Whitehead-Segel equa-tion in rectangular domains. Finite size effects onpatterns are also highly relevant for Turing pat-terns in small systems, as for instance in cells27–30.

a)walter.zimmermann@uni-bayreuth.de

I. INTRODUCTION

Patterns occur spontaneously in a plethora of livingor inanimate driven systems, such as in the atmosphereor in convection cells, in biological cells, in chemical re-actions, or as vegetation patterns, to name just a fewexamples1–15. Already the esthetic appeal of patterns isimmediately apparent to all observers2. Patterns fulfillalso important functions in nature. For example, self-organized patterns in biology guide size sensing31, posi-tioning of protein clusters in the cell center in advanceof cell division29 or in self-driven morphogenesis12. Pat-terns enhance transport in fluid systems1,11 or they arethe basis of successful survival strategies for vegetationin water-limited systems13,14.

Patterns are multistable, i.e. they are stable for dif-ferent wavenumbers within a stability band1,3,18, some-times also beyond a seondary instability32. In quasi-one-dimensional systems, the stability band is boundedby the Eckhaus instability1,18,24–26,33–40, which alsohas its two-dimensional generalization in anisotropicsystems41,42. In two-dimensional isotropic systems, thestability band of stripe patterns is in addition boundedby the zigzag instability1,18,43.

In nature, patterns are always exposed to boundaries,be it the walls of a convection cell, the finite size of a Petridish, or the cytosol bounding membrane for intracellularprocesses. Along these boundaries, the fields describingthe patterns must satisfy certain boundary conditions.Some boundary conditions suppress the pattern near theboundary. In longitudinal direction they act even in longsytems far into the volume and significantly restrict thestable wavenumber band19. In contrast, periodic andno-flux boundary conditions, for example, impose no re-striction on the stable wavenumber band, except that thewavenumbers can take only discrete values. In rectangu-lar systems with amplitude-suppressing boundary condi-tions, stripe patterns prefer an orientation perpendicular

2

to these edges1,44,45.

Patterns are also restricted to finite ranges when thecontrol parameter generating a pattern falls to subcriticalvalues outside a subrange. Examples include photosen-sitive chemical reactions. There, pattern formation canbe suppressed by illuminating the reaction cell outside afinite range46 or even controlled by spatially modulatedillumination47–49. Another example is protein patternsoccurring in finite reactive subdomains of substrates50.In such examples, orientations are often perpendicularto non-resonant control parameter drops45,51. In thecase of steep control parameter decays, stationary stripsmay orient parallel to the boundaries due to resonanceeffects51. In quasi-one-dimensional systems, spatial vari-ations, called ramps, break the translational symmetryand drastically reduce the width of the stable wavenum-ber band21–23,35,52.

No-flux boundary conditions play a central role forreaction-diffusion patterns in cells and elsewhere. No-flux boundary conditions break translational symmetryand fix the phase of a periodic pattern at the boundary,but they share the property of leaving the threshold un-touched and, in medium sized systems, also the stablewavenumber band. Both boundary conditions are verywell suited to study the direct influence of the system sizeon the stability range of striped patterns. From inves-tigations on quasi one-dimensional systems it is knownthat a decrease of the system size leads to an increaseof the Eckhaus stable wavenumber band24, which is alsoconfirmed experimentally25,26. This is an opposite trendas obtained for medium sized and long systems with am-plitude suppressing boundary conditions in longitudinaldirection and for ramps.

Therefore, an interesting question arises: how willthe zigzag instability boundary of the stable wavenum-ber band be shifted by reducing the size of the domaincontaining the patterns? We address this question bystudying the stability of stripe solutions of the Swift-Hohenberg model in rectangular domains as well as andsimultaneously with the universal Newell-Whitehead-Segel equation. Since in rectangular domains stripe pat-terns are oriented perpendicular to edges with amplitudesuppressing conditions, we choose either periodic or no-flux boundary conditions along the short sides of the rect-angle and combine them with these two boundary con-ditions also along the long sides of the rectangle. Thisallows us to analytically determine the Eckhaus stabilityboundary and the zigzag stability boundary in sectionIII. Combining these boundary combinations gives verygood analytical estimates of the shifts of the Eckhaus andzigzag stability boundary with decreasing system size,for several boundary conditions along the transverse di-rection. This is also checked numerically in section IV.There we also compare the analytical results on the in-stability limits with numerical results for the case whenamplitude-suppressing boundary conditions are used inthe transverse direction.

The results in section ,III describe that in small sys-

tems one finds remarkable broadenings of the stablewavenumber band by reducing the system size. Also,stripes in two spatial dimensions behave quasi-one-dimensional in rather narrow systems. In contrast, theevolution of a stripe pattern from an unstable to a sta-ble wavenumber already in medium-size systems dependssignificantly on the nature of the boundary conditions, aswe show with exemplary simulations in Sec. V. A sum-mary and conclusions are given in section VI.

II. MODELS

Generic properties of stripe patterns just above thresh-old can be described by models such as the isotropicSwift-Hohenberg (SH) model1,43,53. It contains the char-acteristic wavenumber q0 of the pattern and allows intwo spatial dimensions also the modeling of spatial vari-ations of the local wavevector of periodic patterns. Es-sential universal properties of periodic patterns are alsocaptured by the dynamics of the pattern envelope varyingslowly on the wavelength of the pattern.1,18,43. The dy-namical equation for the envelope in isotropic systems isdescribed by the so-called universal Newell-White-Segelequation (NWSE)1,17,18,43,54, which can be derived fromthe SH model as well1. We use here both complementarymodel descriptions of stripe patterns.

A. Swift-Hohenberg model

The rotational invariant Swift-Hohenberg model forthe scalar field u(r, t) above supercritical bifurcationsis1,43,53,

∂tu = εu− (q20 +∇2)2u− u3 , (1)

with the control parameter ε and the intrinsic wavenum-ber q0. Near the threshold of periodic patterns (εc = 0)the solutions of the SH-model can be also expressed interms of a slowly varying amplitude A(x, y, t) as follows:

u(x, y, t) = A(x, y, t)eiq0x +A∗(x, y, t)e−iq0x + h.o.t. .(2)

B. Newell-Whitehead-Segel equation

The dynamical equation for the patterns envelopeA(x, y, t) can be derived from the SH model as well asother pattern forming systems, such as from the basicequation of thermal convection in liquids. Through asystematic multiscale perturbations analysis around theonset of periodic patterns and using the property that theenvelope A(x, y, t) varies slowly on the scale of a wave-length of the pattern, 2π/q0,

1,17,18,43,54 one obtains theso-called Newell-Whitehead-Segel equation (NWSE) for

3

the amplitude A(x, y, t):

τ0∂tA = εA+ ξ20(∂x − i

2q0∂2y)

2A− g0|A|2A . (3)

The systems specific properties are covered by the co-efficients, i.e. the values and their meanings dependon the system but the form of the NWSE is universal.If Eq. (3) is derived from the SH equation Eq. (1) oneobtains τ0 = 1, ξ0 = 2q0 and g0 = 3. The univer-sal NWSE with the coefficients corresponding to the SHmodel reproduces stability ranges of stripe pattern nearthe threshold as well, as exemplarily shown in AppendixA4.

C. Boundary conditions

Here, the nonlinear stripe patterns are investigated inrectangular areas with respect to four different boundaryconditions. One type are periodic boundary conditions(PBC)

u(x = 0, y) = u(x = Lx, y) , (x-PBC) (4)

u(x, y = 0) = u(x, y = Ly) . (y-PBC) (5)

PBC take just into account finite-size effects and the pos-sible wavelengths can take only discrete values. We alsoconsider Neumann boundary conditions, which are alsoknown as no-flux boundary conditions:

(~n · ∇)u = 0 = (~n · ∇)3u . (BCI) (6)

Vanishing amplitude and curvature normal to the rect-angular boundary is implemented by he third type ofboundary conditions considered here:

u = 0 = (~n · ∇)2u . (BCII) (7)

The fourth type of boundary conditions for the fieldu(x, y, t),

u(x, y)|y=0,Ly= 0 = ∂y u(x, y)|y=0,Ly

(BCIII) (8)

is only used here along the long sides at y = 0, Ly. Theseboundary conditions are suitable for example for stripepatterns in thermal convection in finite boxes, to modelso-called no-slip boundary conditions for the flow veloc-ity. Effects of the boundary conditions BCIII in bothdirections in a two-dimensional systems are investigatedfor the SH model also in Ref.44.

D. Stationary single-mode solutions of the SH model

Perturbations u = u exp(σt + iq · r) with q = (q, p) ofthe basic state u = 0 of the SH equation grow beyondthe so-called neutral curve

ε0(qn, pm) = (q20 − q2n − p2m)2 . (9)

Note, that in finite systems only discrete values qn and pmmatch into the system for the boundary conditions PBC,BCI and BCII. For PBC one has qn = n2π/Lx and pm =m2π/Ly. For BCI and BCII the discretization steps arehalf the size: qn = nπ/Lx and pm = mπ/Ly. Considerstripe perturbations with periodicity in the x-directionand taking into account the boundary conditions one has:

u = u exp(σt) cos(qnx) , (BCI)

u = u exp(σt) sin(qnx) . (BCII) (10)

The growth rate vanishes along ε0 and beyond ε0(q) theperturbations grow up to a saturation amplitude.With σ = 0 and u = 2As in Eq. (10) and for PBC

or BCI in the y-direction one obtains after projection ofthe SH equation (1) onto cos(qnx + ϕ) with ϕ = 0 forBCI, ϕ = π/2 for BCII and arbitrary ϕ for PBC in thex-direction the expression for the following amplitude ofstripes:

A2s =

[ε− (q20 − q2n)

2]/3 . (11)

The determination of the threshold and the amplitudein the case of BCIII requires essentially a numerical ap-proach as in Ref.44 and the onset of the periodic patterntakes place at higher values of ε. In the case of BCIIin the y-direction and the stripe axis parallel to y thenonlinear solution has to be determined numerically aswell.

III. STABILITY BOUNDARIES OF STRIPES

The linear stability of stripe solutions of the SH modelin rectangular domains can be studied analytically for pe-riodic boundary conditions in Eq. (4), the no-flux bound-ary conditions Eq. (6) and for BCII-type boundary con-ditions in Eq. (7). This is described in this section andalso includes the analytical determination of the Eckhausstability boundary of stripe solutions as well as the zigzagstability boundary for stripes. Solutions of the Newell-Whitehead-Segel equation and their stability are delin-eated in Appendix A1 including a comparison with thefollowing results obtained for the SH model.

A. The zigzag instability for the SH model

To investigate the zigzag instability we add a smallperturbation v(x, y, t) to the stripe solution u0 =As exp(iqnx) + cc: u = u0 + v. A linearization of Eq. (1)with respect to v gives the linear equation,

∂tv = [ε− (q20 +∇2)2 − 3u20]v +O(v2) . (12)

It is solved analytically by the ansatz

v = eσt cos(qnx+ ϕ)(eipjyv1 + e−ipjyv2

)(13)

4

with discrete wavenumbers qn and the following bound-ary types in the x-direction, PBC, BCI (ϕ = 0) and BCII(ϕ = π/2) and discrete values pj in the y-direction forPBC and BCI boundary conditions . The only differencefor the boundary conditions are the allowed discrete val-ues of qn and pj . These are qn = nπ/Lx for BCI, BCIIand qn = 2nπ/Lx for PBC in the x-direction and anal-ogously in the y-direction for BCI pj = jπ/Ly and forPBC pj = j2π/Ly. The result contradicts the recentclaim in Ref.55 that the ansatz for the zigzag instabilityin Eq. (13) does not hold in the case of no-flux bound-ary conditions in the x-direction. Moreover, in Ref.55 itwas claimed, that the perturbation v(x, y, t) must becomesmall near x = 0, Lx, which is definitely not imposed byno-flux boundary conditions BCI.Collecting the linear independent contributions ∝

ei(qnx±pjy) in Eq. (12) gives two coupled homogeneousequations for v1 and v2 with the solubility condition

∣∣∣∣

L − 6A2s −3A2

s

−3A2s L− 6A2

s

∣∣∣∣= 0 , (14)

where the linear operator L is given by

L = ε− σ − (q20 − q2n − p2j)2 . (15)

With the stationary amplitude As(qn) from Eq. (11) thegrowth rate takes the following form:

σ = p2j[2(q20 − q2n

)− p2j

]. (16)

For a nonlinear periodic solution of wavenumber qn thisgrowth rate σ of the perturbation v with the transver-sal wavenumber pj becomes positive when the wavenum-ber qn becomes smaller than qzz at the zigzag stabilityboundary:

qn < qzz =

√

q20 −p2j2

. (17)

The shift of qzz to a value smaller than q0 is deter-mined by the smallest perturbation wavenumber p1 thatmatches into the interval [0, Ly]. p1 is two times larger forperiodic boundary conditions and therefore the shift ofqzz away from q0 is larger in the case of periodic bound-ary conditions, compared to BCI. This also means thatthe stable q range for stripes is stronger enhanced forPBC than for BCI as shown in Fig. 1 below.

B. Finite size effects on the Eckhaus instability

The determination of the stability of periodic patternsagainst longitudinal perturbations gives the so-calledEckhaus-boundary stability boundary and for this a one-dimensional analysis of Eq. (12) is sufficient. In simu-lations one can achieve a quasi one-dimensional behav-ior of stripe patterns by choosing in simulations a small

extension Ly (see also Sec. IV). Therefore, we choose aone-dimensional ansatz v(x, t) for Eq. (12):

v = eσt[eiqnx

(eikjxv1 + e−ikjxv∗2

)+ cc

]. (18)

In finite systems also the wavenumber kj of the pertur-bation is limited to discrete values. For boundary con-ditions BCI and BCII along x one has kj = jπ/Lx andfor periodic boundary conditions one has kj = j2π/Lx.Also the longitudinal instability is a along wavelength in-stability, i.e. growth rate becomes at first positive at thesmallest perturbation wavenumber k1. If the perturba-tion v is growing with a wavenumber k1, then during theinstability process either a node is added to the stripe so-lution or removed. By using the ansatz (18) in Eq. (12)and collecting the contributions ∝ ei(qn±k1)x one obtainstwo coupled homogeneous equations for v1 and v2 withthe solubility condition

∣∣∣∣

L+ − σ −3A2s

−3A2s L− − σ

∣∣∣∣= 0. (19)

and the abbreviation

L± = −ε+ 2(q20 − q2n)2 − (q20 − (qn ± k1)

2)2︸ ︷︷ ︸

M±

. (20)

The growth rate of the perturbation expressed in termsof L± is given by

σ =1

2

(

L+ + L− +√

(L+ − L−)2 + 36A4s

)

. (21)

The neutral stability condition σ = 0 for periodic solutionof wavenumber qn is given by

(L+ + L−)2 = (L+ − L−)

2 + 36A4s (22)

or in different form by

L+L− = 9A4s . (23)

This gives the control parameter value ε(qn) = εn at theneutral stability of a stripe pattern of wavenumber qn

ε(qn) = εn =M+M− − (q20 − q2n)

4

M+ +M− − 2(q20 − q2n)2. (24)

For ε > εn the periodic solution is stable and below inthe range ε0(qn) < ε < εn linear unstable with respectto the longitudinal perturbation of wavenumber k1.

IV. LINEAR STABILITY DIAGRAMS OF STRIPES

Stability diagrams of stripe patterns with theirwavevector q = (q, 0) along the x axis are presented inthis section. The position of the zigzag-stability bound-ary, qzz , and the Eckhaus-stability boundary (E) of stripepatterns in a rectangular domain are shown in Fig. 1 fortwo widths Ly, in the y-direction either periodic (PBC)

5

0.8 0.9 1.0 1.1 1.2

q/q0

0.0

0.2

0.4

N

E PBC

BCIε

FIG. 1. Shown is the neutral curve (solid line), the Eckhaus-stability boundary (dashed line) and the vertical lines markthe zigzag instability qzz given by Eq. (17) for different Ly

and for BCI and PBC in the y-direction. The vertical dottedline marks qzz for both, BCI and PBC at Ly = 40λ0. ForLy = 2λ0 the position of qzz is stronger shifted for PBC (leftdash-dotted line) than for BCI (right dash-dotted line).

or no-flux boundary conditions (BCI) and no-flux bound-ary conditions at x = 0, Lx. The solid line N in Fig. 1 isthe neutral curve described by Eq. (9).The vertical dotted line in Fig. 1 marks the position of

the zigzag-stability boundaries qzz obtained via Eq. (17)for a broad rectangle with Ly = 40λ0 and for BCI andPBC at y = 0, Ly. The qzz for both cases is indistin-guishable. In a narrow system with Ly = 2λ0 the zigzag-stability boundary for PBC in the y-direction in Fig. 1 isabout four times as far shifted to the left than for BCI.The reason, the smallest wavenumber of a perturbationof the periodic stripe solution in a system of width Ly isfor PBC twice as large as for BCI, i.e., pPBC

1 = 2pBCI1

and its square p21 contributes to qzz in Eq. (17). Thus,the location of the zigzag stability boundary depends es-sentially on the system extent Ly and on the boundaryconditions in the y-direction of the stripe axis, but noton the boundary conditions perpendicular to the stripewavevector in the x-direction.In one-dimensional systems with large Lx the Eckhaus-

stability boundaries (ESB) for BCI and PBC in the x-direction are indistinguishable and given by the dashedline in Fig. 2. The location of the ESB depends on thelength Lx and also on boundary conditions in the x-direction. To illustrate this, we show in Fig. 2 also theEckhaus boundary for a short system with only two pe-riodic pattern units in the system. For this purpose, weconsider the case of a pattern of wavenumber q in a sys-tem of suitable length Lx = 2(2π/q) for the two boundaryconditions PBC and BCI at x = 0, Lx. The ESB is de-termined by Eq. (24), but with k1 = π/Lx for BCI andk1 = 2π/Lx for PBC. For both short systems, the ESBintersects the neutral curve, as shown in Fig. 2. For εbelow this intersection the periodic patterns are stablefor all q between neutral curve, i.e. for all stripe patternsthat emerge. The dependence of the Eckhaus boundaryon the system length was already recognized in Ref.24

and this length dependence was also confirmed in experi-

0.7 1.0 1.3

q/q0

0.0

0.2

0.4 N PBC

BCI

ε

FIG. 2. Shown is the ESB in the one-dimensional case forboth, PBC and BCI in the x-direction. In a long system withLx = 40λ0 the ESB’s are indistinguishable for PBC and BCI(dashed line). In a short system with Lx = 2λ0 (dash-dottedlines) the Eckhaus stability range is broader for PBC than forBCI. The solid line N is the neutral curve.

ments on Taylor vortex flow in Ref.26. The ESB is similaras for the zigzag boundary for PBC shifted further fromthe Eckhaus boundary for very long systems than for theboundary condition BCI. The difference is again causedby the different value of the perturbations wavenumber,similar as for the zigzag instability.The wavenumber along the Eckhaus curve, as for ex-

ample along the dashed curve in Fig. 2, we call qE(ε).Its deviation from the preferred wavenumber q0 is QE =qE − q0. Defining analogous qN (ε) as the wavenumberalong the neutral curve and its deviation from the pre-ferred wavenumberQN = qN−q0, the ratio between bothdeviations follow near the threshold (ε ≪ 1) the universallaw of stripe patterns in one spatial dimension18,24,43:

QE

QN

=1√3. (25)

The ratio becomes larger in systems of finite extensionLx. In addition, the ratio depends on ε, Lx and on theboundary conditions in the longitudinal x-direction. Wedetermine via Eq. (24) the wavevector along the Eckhausboundary, qE(ε, k1), as function of ε with k1 = π/Lx

for BCI (resp. k1 = 2π/Lx for PBC). The ratio betweenQE(ε, k1) and QN = qN−q0 is shown for the SH model inFig. 3 as function of Lx for two different values of ε andtwo boundary conditions. For an infinite system nearthreshold this ratio is given by Eq. (25), which is indi-cated by the dotted horizontal line in Fig. 3. For PBCthe ESB is shifted further away from the value for infi-nite systems in Eq. (25) and closer to the neutral curveε0(q) than for BCI. This means the ESB reaches the neu-tral curve (see also Fig. 2) and thus the largest possiblevalue QE/QN = 1 for a chosen ε already at larger systemlengths Lx than for no-flux boundary conditions BCI.The value qzz at the ε-independent zigzag-stability

boundary has different values for different boundary con-ditions in the y-direction as indicated in Fig. 1. The de-pendence of qzz on the system width Ly is shown in Fig. 4for the two boundary conditions PBC and BCI in the y-

6

0 10 20 30 40

Lx/λ0

0.6

0.8

1.0Q

E/Q

NPCB

BCI

PCB

BCI

FIG. 3. Shown is the ratio QE/QN between the half-widthof the Eckhaus-stability band, QE , and the half-width of theneutral curve, QN , as function of the system length Lx/λ0

for the two boundary conditions PBC and BCI and for thecontrol parameter ε = 0.01 (dashed lines) and ε = 0.4 (solidcurves).

1 3 5 7

Ly/λ0

0.7

0.8

0.9

1.0

qzz

FIG. 4. The zigzag stability boarder qzz, given by Eq. (17), isshown as a function of the system width Ly for no-flux bound-ary conditions (BCI) (solid line) and for periodic boundaryconditions (dashed line) in the y-direction.

direction. qzz decreases with decreasing Ly in both cases,but stronger for PBC. The reason is again the larger per-turbation wavenumber k1 in Eq. (20) for PBC.A consequence of the Ly dependence of qzz in Fig. 4

is further illustrated in Fig. 5. Shown is the neutralcuve (dashed-dotted), the Eckhaus boundary (solid line)and the zigzag instability for the two lateral extensionsLy = 2λ0, 1.5λ0 and three different boundary conditionsin the y-direction. For the wider system with Ly = 2λ0

the zigzag instability qzz for BCII (+ symbols) is lo-cated between qzz for BCI (vertical solid line) and qzzfor PBC (vertical dashed line). Also for the narrowersystem with Ly = 1.5λ0 the zigzag instability qzz for thetype BCII boundary condition (× symbols) is betweenqzz for BCI boundary conditions (vertical dashed-dottedline) and qzz for PBC (dashed-dot-dotted line). Whilethe results for BCI and PBC are determined by the ex-pression in Eq. (17), the zigzag instability for y-BCII aredetermined via simulations. In this sense the analyticalformula of qzz for BCI and PBC gives a reasonable es-timate about the location of the zigzag boundaries forfurther boundary conditions in the y-direction.

FIG. 5. In a long system with Lx = 40λ0, the ESB (solidline) and the zigzag stability lines are shown for differentboundary conditions in the y-direction and for two widthsLy = 2λ0, 1.5λ0. Width Ly = 2λ0 (red): The vertical solidline marks the zigzag stability boundary qzz for BCI, the ver-tical dashed line qzz for PBC and the +-symbols mark qzzfor BCII. Width Lx = 1.5λ0 (blue): The vertical dash-dottedline gives qzz for BCI, dash-dot-dotted line for PBC and thecrosses × mark qzz for BCII.

In Fig. 5 the zigzag boundary crosses for differentboundary conditions the Eckhaus-boundary (solid curve)at different values of ε. For control parameter valuesbelow this intersections of the zigzag and the Eckhausboundary the systems behaves quasi-one dimensional.I.e. below these ε the zigzag (ZZ) instability becomesirrelevant, because the Eckhaus instability sets in earlier.Since the ESB depends for small system lengths Lx on

the boundary condition in the x-direction, the crossing ofthe ZZ and the Eckhaus stability boundary depends onthe boundary conditions in the x and y direction. There-fore, the transition to a quasi one-dimensional behaviorof stripe patterns depends on the control parameter ε,the system size and the used boundary condition in eachdirection. This is shown in Fig. 6 for Lx = 3λ0, where thefirst part of the curve label refers to the boundary condi-tion in the x-direction and the second part refers to theboundary condition in the y-direction. Below these fourcurves in Fig. 6 the zigzag instability becomes irrelevantfor these systems sizes and boundary conditions and thestripes behave one dimensional. Conversely, the zigzaginstability limits above these curves in Fig. 6 the stabilitystripe pattern in the range q < q0 for all the consideredsystems sizes and boundary conditions considered in thiswork, which is in contrast to Reference55.

V. EXAMPLES OF THE NONLINEAR EVOLUTION OFUNSTABLE STRIPES IN FINITE SYSTEMS

In this section, we exemplify how stripe patterns evolveafter applying small perturbations from an unstablewavenumber q < q0 through nothing but a zigzag in-stability to stripe patterns at a stable wavenumber. Weuse simulations of the SH model (1) in a rectangular do-main with Lx = 40λ0 and Ly = 20λ0 and choose severaldifferent boundary conditions along the sides of the rect-

7

0 1 2 3

Ly/λ0

0.0

0.1

0.2

ε

PBC-BCI

BCI-BCI

PBC-PBCBCI-PBC

FIG. 6. For a short system length Lx = 3λ0 and differentcombinations of BCI and PBC in the x- and the y-direction,the ε position of the intersections between the ESB and thezigzag boundary is shown as function of the system width Ly .The first part of the curve designation describes the boundaryin the x-direction.

angle.

In Fig. 7 and Fig. 8 we started simulations in the rect-angular domain by using a periodic initial pattern withε = 0.4 and a wavenumber q = 0.9 which produce 36stripes in the system. In Fig. 7 no-flux boundary con-ditions (BCI) are used along all four sides, whereas inFig. 8 we replaced BCI by periodic boundary conditionsin the x-direction. For the chosen system extensionsthe zigzag-stability boundaries is in both cases nearlyindistinguishable at qzz . q0 as shown in Fig. 1, i.e.the starting wavenumber q = 0.9 is far in the unstablerange. While the two different boundary conditions leavethe zigzag-stability boundary nearly untouched for largerLy = 20λ0, the temporal evolution of the stripe patternsfrom an unstable to a stable wavenumber q differs. Awayfrom the boundaries at x = 0, Lx the unstable stripeswith q0 = 0.9 become according to the zigzag instabilityundulated, as can be seen in Fig. 7b) and Fig. 8b). Forperiodic boundary conditions at x = 0, Lx these stripeundulations occur also at the boundaries as shown inFig. 8c). No-flux boundary conditions fix the phase of thestripe pattern with its maximum or minimum u(x, y, t)at x = 0, Lx. Therefore no undulations evolve at andnear the boundaries, as indicated in Fig. 7c). Accord-ing to the BCI induced constraint on the phase near theboundaries one has for BCI at x = 0, Lx a shorter tran-sient time to reach finally a stable straight stripe patternwith q0 . q as in indicated by Fig. 7d) and Fig. 8d). Inboth cases one ends up with a state composed of 40 pat-tern units, i.e. with q0 . q. When we start simulationsfor both boundary conditions with an initial solution ofwavenumber q = 0.975, which corresponds to 39 patternunits in the system, the pattern evolves again to a stateof 40 units with q0 . q.

In Fig. 9 and Fig. 10 we started simulations in a rectan-gular area with the same initial wavenumber q = 0.9 andε as in Fig. 7. However, we replaced at y = 0, Ly no-fluxboundary conditions (BCI) by the boundary conditionsof type BCII in Fig. 9 and by type BCIII boundary con-

FIG. 7. Shown are four snapshots of a simulation of the SHmodel in a rectangular domain of the side lengths Lx = 40λ0

and Ly = 20λ0; color scale represents the minimal (blue) andthe maximal (red) values of u; no-flux boundary conditions(BCI) along the four boundaries and at ε = 0.4. In (a) theinitial solution with the wavenumber qs = 0.9 at time t = 0is shown, which is undulated by the zigzag instability in (b)at t = 103. The resulting defects in (c) at t = 7.5 · 103 healrapidly due to the fixed phases at the edges and result in thesolution with wavenumber q = 1.0 at t = 66 · 103 in (d).

dition in Fig. 10. As indicated in Fig. 5, the position ofthe zigzag-stability boundary qzz is in the case of BCIIboundary conditions at y = 0, Ly stronger influenced inrather narrow systems than for no-flux boundary condi-tions, which is also true for BCIII boundary conditions.However, the starting wavenumber qs = 0.9 is

8

FIG. 8. Shown are four snapshots of a simulation of the SHmodel in a rectangle and the same parameters as in Fig. 7.Here, the no-flux boundary conditions are replaced by peri-odic boundary conditions (PBC) in the x-direction: startingfrom a solution with qs = 0.9 in (a) at time t = 0, the so-lution becomes modulated at t = 2 · 103 in (b). The defectsappearing at t = 3.5 · 103 in (c) are eliminated during furtherevolution and result in a solution with q = 1.0 at t = 171 ·103 .

again considerably below qzz of PBC, which causes thestrongest shift. One can recognize in Fig. 9 and Fig. 10that the boundary conditions BCII and BCIII suppressstripe pattern close to y = 0, Ly. Also for these bound-ary conditions at y0 = 0, Ly the unstable stripe patternat qs = 0.9 becomes undulated in the bulk via the zigzaginstability as indicated in Fig. 9b) and Fig. 10b). The

FIG. 9. Shown are four snapshots of a simulation in a rect-angle for the same parameters as in Fig. 7, but with type IIboundary conditions (BCII) in the y-direction: a) Initial so-lution with wavenumber qs = 0.9 at t = 10, b) t = 500, c)t = 5.4 · 103 and d) t = 109 · 103 a stable solution with 39periods and q = 0.975.

further evolution is slightly different from the evolutionshown in Fig. 7. The major difference is that at a similarsimulations time the state in Fig. 9d) is composed of 39periodic units and in Fig. 10d) by 38 periodic units.

9

FIG. 10. Shown are four snapshots of a simulation in a rectan-gle for the same parameters as in Fig. 7, but type III boundaryconditions (BCIII) in the y-direction: a) Initial solution withwavenumber qs = 0.9 at t = 10, b) t = 500 and c) t = 5 · 103

and in d) a solution with 38 stripes and q = 0.95 so far.

VI. SUMMARY AND CONCLUSIONS

We investigated finite sizes effects on the multistabilityof supercritical bifurcating stripe patterns in rectangulardomains using the generic Swift-Hohenberg model andthe universal Newell-Whitehead-Segel equation. In two-dimensional extended isotropic systems, the wavenumberrange of stable periodic patterns is limited by the longitu-dinal Eckhaus instability and the transverse zigzag insta-bility. We show analytically and numerically for different

combinations of boundary conditions along the edges ofa rectangular domain that the range of wavenumbers forstable stripes increases with a reduction of the systemsize.

Note, also in finite systems the Eckhaus and zigzaginstabilities remain the instabilities limiting the stablewavenumber range of stripe pattern for different bound-ary conditions. The zigzag instability remains the pri-mary transversal instability also for no-flux boundaryconditions in the longitudinal direction of stripe pattern.It is not replaced by another primary instability as re-cently claimed in Ref.55.

The enlargement of the stable wavenumber range ofstripe patterns by the system size reduction is basedon the following insights. The Eckhaus and zigzag in-stabilities are long-wavelength instabilities. By decreas-ing the system size, their destabilizing long-wavelengthmodes are increasingly suppressed. For periodic bound-ary conditions, an entire wavelength of a destabilizingmode must fit into the system, while for no-flux bound-ary conditions, for example, only half a wavelength ofthe destabilizing mode must fit into the finite system.That is, the smallest wavenumber k of the perturbationis twice as large for periodic boundary conditions as forno-flux boundary conditions. According to our analyticalresults, the zigzag stability boundary is shifted propor-tionally to k2. This means, for periodic boundary con-ditions in transverse direction, a reduction of the rect-angle width shifts the stability boundary up to a factorof four more and increases the stable wavenumber rangethan for other boundary condition in transverse direc-tion. The enlargement trend of the stable wavenumberrange is similar for a reduction of the system length inlongitudinal direction by shifting the Eckhaus boundary.

If boundary conditions that suppress the amplitude ofthe stripe pattern are taken in the transverse direction,the numerical results for the zigzag instability boundarylie between the analytical results for periodic and no-flux boundary conditions. This underlines the value ofthe presented analytical results also as an estimate forthe location of the zigzag instability for other boundaryconditions.

By reducing the system width sufficiently, the zigzaginstability limit shifts to smaller values of the wave num-ber than the lower Eckhaus stability limit for not toshort systems. In this case, the zigzag instability is sup-pressed by the longitudinal instability that occurred pre-viously. Below such system widths, the stripe patternbehaves quasi-one-dimensionally in two-dimensional sys-tems. Again, the transition to quasi-one-dimensional be-havior for periodic boundary conditions in the transversedirection occurs at already larger widths than for no-fluxboundary conditions. Also for the transition to quasi-one-dimensional behavior, the analytical results for pe-riodic and no-flux boundary conditions give a good esti-mate for the transition to quasi-one-dimensional behaviorfor other boundary conditions in transversal direction.

In the spatiotemporal evolution from a periodic stripe

10

pattern with a wavenumber below the zigzag instabilitylimit to a stripe pattern with a stable wavenumber, theinfluence of the boundary conditions is already noticeablefor medium-sized systems with about 20 periodic stripes.This is shown for various combinations of boundary con-ditions along the rectangular domain in Section V. Asthese simulations show, in all cases the zigzag instabilityis the destabilizing mechanism, in contrast to the descrip-tion in Ref.55.

The results of this work give also an estimate belowwhich system lengths and widths, for given values of thecontrol parameter, any emerging periodic pattern is alsostable. These insights are important in investigationsof e.g. Turing patterns in very small systems such ascells29. The here derived generic limitation of the stablewavenumber bands addresses also the so called robust-ness problem29,56.

ACKNOWLEDGMENTS

Support by the Elite Study ProgramBiological Physicsis gratefully acknowledged.

DATA AVAILABILITY

The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.

Appendix A: Newell-White-Segel equation (NWSE) andstability of stripes

For completeness we include also the stability bound-aries of stripes determined via the NWSE in Eq. (3) forno-flux boundary conditoins and periodic boundary con-ditions, which we can compare with the results of the SHmodel.

1. Linear stability of stripes within the NWSE

The NWSE has the stationary solutions

A = FneiQnx (A1)

with the wavenumber Qn = qn − q0 and the amplitudeF 2n = (ε − ξ20Q

2n)/g0. One has F 2

n > 0 for ε > ε0 andFn = 0 along the neutral curve:

ε0 = ξ20Q2N . (A2)

2. Zigzag instability within the NWSE

At first we investigate the zigzag instability of thestripe solution. For this we use the ansatz A = (F +v(y))eiQnx with a small y-dependent perturbation v(y, t).By neglecting higher order terms of v in Eq. (3), a linearequation for v results:

τ0∂tv = εv + ξ20

[

iQn − i

2q0∂2y

]2

v − g0F2v − F 2v∗.

(A3)

This linear equation may be solved by

v = eσt[eiLjyv1 + e−iLjyv∗2

], (A4)

where the wavenumber for no-flux boundary conditionsat y = 0, Ly is Lj = jπ/Ly and for PBC Lj = j2π/Ly.Collecting the contributions ∝ e±iLjy gives two coupledhomogeneous equations for v1 and v2 with the solubilitycondition

∣∣∣∣

L − στ0 −g0F2

−g0F2 L− στ0

∣∣∣∣= 0 (A5)

and the abbreviation

L = −ε− 2ξ20Q2n − ξ20

(

Qn +L2j

2q0

)2

. (A6)

Herein the growth rate of the perturbation is

στ0 = −L2j

2q0

(

2Qn +L2j

2q0

)

. (A7)

The stripe solution with the amplitude given by Eq. (A1)is stable with respect to the perturbation in Eq. (A4) inthe range

Qn > −L2j

4q0= Qzz. (A8)

3. Eckhaus-instability of stripes

Next we investigate the stability of stripe solutionswith respect to longitudinal perturbations in long a quasione-dimensional systems, i.e. with a small width Ly =π/2q0.In this case we investigate with Eq. (A3) the dynamics

of perturbations v(x, t) with respect to the solution givenby Eq. (A1). We solve the linear equation (A3) with thefollowing ansatz

v = eσt(eiKlxv1 + e−iKlxv∗2

), (A9)

and with the wavenumber Kl = lπ/Lx. If the perturba-tion v is growing with a wave numberK1, then during the

11

instability process either nodes are added to the stripe so-lution, cf. Eq. (A1), or removed. With the ansatz (A9) inEq. (A3) and collecting the contributions ∝ e±iKjx, givestwo coupled homogeneous equations for v1 and v2 withthe solubility condition

∣∣∣∣

L+ − στ0 −g0F2n

−g0F2n L− − στ0

∣∣∣∣= 0 (A10)

and

L± = −ε+ 2ξ20Q2n − ξ20(Q

2n ±K2

l )︸ ︷︷ ︸

M±

. (A11)

The growth rate of the perturbation expressed in termsof L± is given by

σ =1

2

(

L+ + L− +√

(L+ − L−)2 + 4g20F4n

)

. (A12)

The neutral stability σ = 0 condition for the n-node so-lution supplies

L+L− = g20F4n . (A13)

This gives the stability boundary εn of the n-node solu-tions in the εn − qn plane

εn =M+M− − ξ40Q

4n

M+ +M− − 2ξ20Q2n

. (A14)

The n-node solution is stable (unstable) above (below)this curve.

4. Comparison of the NWSE to the SH model

As mentioned above the NWSE can derived from theSH model for the coefficients τ0 = 1, ξ0 = 2q0 and g0 = 3by a weakly nonlinear analysis1. This approximationholds near the threshold, where the neutral curve of theSH model Eq. (9) becomes also parabolic similar as inEq. (A2). For higher values of the control parameter ε,this two curves N for the SH and the NWSE differ ascan be seen in Fig. 11. In contrast to this difference be-tween the neutral curves of the NWSE (dashed lines) andthe SH model (solid lines) the Eckhaus stability bound-aries (E) is nearly identical. The zigzag instability (Z)even is indistinguishable for the two models. Thereforethe qualitative and quantitative results of this section,are with universal character since the universality of theamplitude equation.

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