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Origin of the Pinning of Drifting Monostable Patterns

M. G. Clerc,

1

C. Fernandez-Oto,

1

M. A. Garcı

´a-N

˜ustes,

1

and E. Louvergneaux

2

1

Departamento de Fı

´sica, Facultad de Ciencias Fı

´sicas y Matematicas, Universidad de Chile, Casilla 487-3, Santiago, Chile

2

Laboratoire de Physique des Lasers, Atomes et Mole

´cules, UMR CNRS 8523, Universite

´Lille 1,

59655 Villeneuve d’Ascq Ce

´dex, France

(Received 10 April 2012; revised manuscript received 29 June 2012; published 5 September 2012)

Under drift forces, a monostable pattern propagates. However, examples of nonpropagative dynamics

have been observed. We show that the origin of this pinning effect comes from the coupling between the

slow scale of the envelope to the fast scale of the modulation of the underlying pattern. We evidence that

this effect stems from spatial inhomogeneities in the system. Experiments and numerics on drifting

pattern-forming systems subjected to inhomogeneous spatial pumping or boundary conditions conﬁrm

this origin of pinning dynamics.

DOI: 10.1103/PhysRevLett.109.104101 PACS numbers: 05.45.a, 42.65.Sf, 47.54.r

Pattern formation far from equilibrium occurs in all

domains of sciences through the spontaneous symmetry

breaking of a ground state [1]. Structures, generated at the

ﬁrst threshold of spatial instability, are generally stationary

and can be either of (i) localized or of (ii) extended-

periodic type. In some particular cases, such patterns con-

tinuously drift at their onset, for instance, when the system

either possess (i) multiple coexisting states, (ii) a Hopf

bifurcation or (iii) is subjected to cross advection or ﬂow.

In the ﬁrst context, spatially localized interfaces connect-

ing different states propagate due to the interplay between

state energies. In the second one, a spontaneous parity

breaking instability occurs that produces steadily drifting

patterns [2,3]. In the third one, the induced cross convec-

tion forces the global patterns to drift [4,5]. However, in all

situations, it has been observed that the structure could

remain motionless or pinned [6–16].

In multistable systems, this phenomenon is called the

‘‘pinning effect’’ and was envisaged by Y. Pomeau more

than 20 years ago [17,18]. This is the case for, e.g., in an

homogeneous and a patterned state [19,20] or else two

patterned states [21]. Pinning effect is a result of the

competition between different energetic states that pro-

duces front propagation and spatial modulations that tend

to block the motion by introducing periodic potential bar-

riers in the dynamics of the front core [7,15]. Depending

on the dominant effect, the front can stay motionless

(locked) over a region of the parameter, called the pinning

range. Above a critical value of the control parameter, the

pinning-depinning transition occurs and the localized pat-

tern (front or domain wall) propagates with periodic leaps.

Increasing further the control parameter, the velocity of the

interface becomes constant in space and time.

The same phenomenon is encountered in drifting mono-

stable systems where single patterns are propagating

[6,9,13,14]. In the latter, the pinning effect is also present.

Theoretical works on pinning behavior have discussed the

effects of the spontaneous translation symmetry breaking

for a second order transition system [22] or else ‘‘nonadia-

baticity’’ for a ﬁrst order transition system [7]. However, no

general framework has been developed to elucidate the

underlying locking mechanism in the class of monostable

systems. Hence, a theoretical work is required for drifting

monostable pattern systems that can be uniﬁed with the

pinning theory developed in multistable systems.

In the present Letter, we show theoretically, numerically

and experimentally that pinning-depinning transitions in-

duced by spatial inhomogeneities in monostable systems

come from the coupling between the small scale of the

pattern modulation and the large scale of its amplitude

envelope. This coupling, which is in contradiction with

the standard multiple scale development assumption, ap-

pears as nonresonant terms in the amplitude equation. The

analytical averaged phase velocity of the pattern agrees

quite well with experimental dynamics of drifting patterns

in a convective Kerr optical feedback system subject to a

Gaussian transverse inhomogeneity.

In one-dimensional spatially extended convective

systems, a well established pattern formed after a ﬁrst

instability threshold drifts as a consequence of a spatial

asymmetric nonlocal interaction. A prototype model used

to describe this effect is the drifting Swift-Hohenberg (SH)

equation [23],

@tu¼"u u3ð@xx þq2Þ2uþ@xuþu2;(1)

where uðx; tÞis a scalar ﬁeld, "is the bifurcation parameter,

qis the pattern wave number, accounts for drift source of

the pattern and is the nonlinear response coefﬁcient.

The SH model was introduced to describe the onset of

Rayleigh-Benard convection; however, recent generaliza-

tions have been used intensively to account for pattern

formation in several systems [23]. Equation (1) describes

a supercritical bifurcation where the variable and parame-

ters scale as u"1=2,qOð1Þ,@x"1=2, and @t"

where "1.For"<0, the system presents a stable

uniform state uðx; tÞ¼0.At"¼0the system bifurcates,

PRL 109, 104101 (2012) PHYSICAL REVIEW LETTERS week ending

7 SEPTEMBER 2012

0031-9007=12=109(10)=104101(5) 104101-1 Ó2012 American Physical Society

the uniform solution becomes unstable, giving rise to

pattern formation. For ">0, the pattern amplitude, at

wave number kc¼q, grows as the square root of ".

To reveal the pinning—depinning phenomenon in the

SH model, we performed numerical simulations with two

different boundary conditions: (i) Neumann (@xu¼0) and

(ii) periodic. Figure 1displays the pattern mean speed hvi

for different values of the drift parameter . Neumann

boundary conditions (black dots) impose a strong spatial

variation of the amplitude close to borders [Fig. 1(a)].

Remarkably, under these conditions, the system exhibits

a pinning range. Within it, the drifting pattern is pushed to

one side reaching a stationary state after a transient state.

Figure 1(b) shows the ﬁnal steady state for this case. Just

above the pinning-depinning transition, the pattern moves

with periodic leaps [Figs. 1(a) and 1(c)]. On the other hand,

for periodic boundary conditions, the system displays a

constant envelope over the whole space. The insets of

Figs. 1(d) and 1(e) show this feature. For this case, the

pattern drifts with nonzero velocity for any value of

parameter 0(dotted line in Fig. 1). The system moves

with almost a constant speed.

It is important to note that spatial inhomogeneities

of a parameter lead also to a pinning-depinning effect.

Numerical simulations performed with a Gaussian

variation of the parameter ", i.e., "¼"0þa0eðxx0Þ=22,

display a similar behavior to those observed for the

Neumann boundary condition. Indeed, the Gaussian spatial

dependence of "imposes a smooth amplitude variation at

the borders inducing a boundary conditionlike.

It is clear from these results that boundary conditions

induce spatial variations for the pattern envelope even

comparable to its modulation amplitude. As a conse-

quence, the system displays a pattern envelope-modulation

coupling close to borders leading to the pinning-depinning

of the front as demonstrated in spatially modulated

media [19–21].

To assess the above proposition, let us consider the

amplitude equation approach of Eq. (1). Close to the

bifurcation ("1), using the ansatz: uðx; tÞ¼

AðyOðﬃﬃﬃ

"

pÞ;tÞexpðikcxÞ=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

32

pþc:c:in Eq. (1) and

imposing a solvability condition to calculate nonlinear

corrections [23], we get

@tA¼"A jAj2AþiA þﬃﬃﬃ

"

p@yAþ"@yyA; (2)

where q stands for the phase velocity at q,

=2qaccounts for the group velocity, and considering an

appropriate spatial scaling. This conventional convective

amplitude equation does not exhibit any phase velocity

locking behavior [18]. Indeed, the amplitude equation ap-

proach is based on the separation of spatial evolution scales

[24]. Such separation becomes relevant in the solvability

condition which implies an inner product of the form,

hfjgi¼ð ﬃﬃﬃ

"

pkc=2ÞRyþ2ﬃﬃ

"

p=kc

yfðy; x= ﬃﬃﬃ

"

pÞgðy; x= ﬃﬃﬃ

"

pÞdx

where yrefers to the slow scale (amplitude), xis the fast

scale (patten), and ff; ggare periodic functions in x.

Considering ﬃﬃﬃ

"

p!0, the spatial variation of the pattern

envelope is slow enough with respect to the underlying

pattern modulation wavelength (@xAkcA= ﬃﬃﬃ

"

p). In the

case of Neumann boundary conditions, however, the cou-

pling between the pattern envelope and its modulation close

to the borders breaks the validity of this scale separation

assumption which turns out to be no longer satisﬁed [18].

Following the asymptotic expansion of the Laplace integral

[25], we obtain the corrective—nonresonant—terms to the

amplitude equation [Eq. (2)] in the limit ﬃﬃﬃ

"

p!0[21,25].

The resulting amended amplitude equation then reads,

@tA¼"A jAj2AþiA þﬃﬃﬃ

"

p@yAþ"@yyA

i ﬃﬃﬃ

"

p

3½3@yA2eikcy= ﬃﬃ

"

p

3@yjAj2eikcy= ﬃﬃ

"

p@y

A2e3ikcy= ﬃﬃ

"

p;(3)

where ¼= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

32

pq2. Notice that nonresonant

terms—the three last terms of Eq. (3)—become relevant

when the spatial derivative of the envelope is no longer

negligible, e.g., close to borders. This leads to a coupling

between the envelope and the spatial modulation of the

underlying pattern. These terms appear as soon as the

system has ﬁnite transverse size (boundary conditions) or

) c)a

−5 0 5

−4

−3

−2

−1

0

1

2

3

4b)

d)

x 10 -3

x 10-3

v

γ

Spacepaceac

TTimeimee

e)

FIG. 1 (color online). Numerical pattern phase velocity hvi

versus drifting parameter (black dots) from Eq. (1). The blue

solid line and red dashed line correspond to the velocity for

Neumann boundary conditions [ﬁtting obtain using Eq. (6)] and

periodic conditions, respectively. Insets: Spatiotemporal pattern

evolution is shown. A schematic image (zoom) of the pattern and

the envelope close to the borders are also included (solid line).

(a)–(c) Null ﬂux boundary conditions and (d), (e) periodic bound-

ary conditions. System parameters are "¼0:09,q¼0:5,¼1

and (a) ¼3:2103, (b) ¼2103, (c) ¼3:2103,

(d) ¼1103, (e) ¼1103.

PRL 109, 104101 (2012) PHYSICAL REVIEW LETTERS week ending

7 SEPTEMBER 2012

104101-2

possesses inhomogeneous parameters, which is quite often

the case in reality.

To emphasize the effects of the corrective terms,

we performed numerical simulations of Eq. (3). We con-

sidered two different boundary conditions: (i) Neumann

condition, @xA¼0and (ii) Dirichlet condition, A¼0.

Measuring the time evolution of the spatially averaged

phase dhðx; tÞix=dt, where hðx;tÞix1=LRL

0ðx;tÞdx,

we get the averaged phase speed hviof the pattern.

Figure 2, summarizes our numerical results for a system

size of L¼600. For Neumman boundary conditions, the

amplitude envelope is homogeneous in space. Then, non-

resonant terms are equal to zero. The averaged phase speed

hviincreases linearly with the drift parameter (Fig. 2, red

dashed line). On the other hand, for Dirichlet conditions,

we obtain large amplitude variations as Agoes to zero at

the borders, so that terms such as @xA2(i.e., nonresonant

terms) play a prominent role. There exists a pinning region

over a large interval of values of . Figure 2(b) shows the

spatiotemporal diagram of the reconstructed ﬁeld uðx; tÞ

from Aðx; tÞ. It is clear that pattern remains stationary even

for nonzero values of . Above a critical value c, the

pattern drifts with periodical leaps (Fig. 2(c)).

To analytically understand the above dynamical behav-

ior, we derive an explicit expression for the pattern speed

hvias a function of parameter . Let us consider the polar

representation Aðx; tÞ¼Rðx; tÞeiðx;tÞ, where Rðx; tÞand

ðx; tÞare the envelope modulus and phase, respectively.

Replacing the polar representation in Eq. (3), and taking

the imaginary part, we obtain,

R@t¼R@xxþ2R@x@xRþR þR@x

þ2R@xRcosðkcxþÞ2R2@xsinðkcxþÞ

2R@xRsinðkcxÞ:(4)

The real part determines the dynamics of Rðx; tÞwhich, at

dominant order, is stationary and independent of .As

depicted in Fig. 3,ðx; tÞis composed of two different

superimposed temporal dynamic behaviors: (i) a periodic-

like and (ii) a monotonically increasing one. Based on this

observation, we propose the following ansatz: ðx; tÞ¼

!ðx; tÞþ

c

ðtÞwhere !and

c

, respectively, account

for the periodic (with dominant frequency !) and the

linear dynamics. Averaging Eq. (4) on space, we obtain

a time dependent only expression. Next, let us take as the

time average hðx; t0Þit!

TRtþT

tðx; t0Þdt0. Denoting

hhfðx; tÞixithfðx; tÞix;t, it is clear that the terms of the

form hR@xRFð!Þix;t where Fð!Þis a periodic function,

are equal to zero from periodicity arguments. Meanwhile,

terms of the form hR2@x!Fð!Þix;t remain. Close to the

pinninig-depinning transition, dhix=dt has a slow dynam-

ics. Therefore, the slope of

c

ðtÞremains invariant on time,

i.e.,

c

ðtÞﬃh

c

ðt0Þit. Subsequently, using trigonometric

relations, Eq. (4) reads

_

c

ðtÞ¼2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A2

0þB2

0

qcosð

c

þÞ;(5)

where A0¼hR2@x!sinðkxþ!Þix;t =hRðxÞix;t ,B0¼

hR2@x!cosðkxþ!Þix;t=hRðxÞix;t, and cosðÞ¼B0=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A2

0þB2

0

q. Equation (5) can be interpreted as an over-

damping particle under the inﬂuence of a periodical and

constant force. Solving analytically Eq. (5) we get the

following average speed [19]:

−8−6−4−202468

−8

−6

−4

−2

0

2

4

6

8

b)

Space

a

T

i

me

c)

a)

<v> -3

-4

β

x 10

x 10

FIG. 2 (color online). Numerical computed pattern phase veloc-

ity hviversus drifting parameter from Eq. (3) for Dirichlet (black

dots) and periodic conditions (red dashed line). Curve ﬁtting (blue

solid line) with Eq. (6), c¼4:5104. Insets: Spatiotemporal

pattern evolution of a reconstructed ﬁeld uðx; tÞfrom Aðx; tÞwith

"¼0:4,¼0:1,L¼600, and (a) ¼7:0104,

(b) ¼3:0104, and (c) ¼7:0104.

0

200

400

0

Time

0

2

-2

4000

Space

φω (x,t)

φ (x,t)

1000

3000

5000

0

0

200

400

600

0

Time

20

Space

FIG. 3 (color online). Spatiotemporal diagram for ðx; tÞwith

"¼0:4,¼0:1,kc¼0:1and ¼7:0104. Inset:

Spatiotemporal diagram for !ðx; tÞ.

PRL 109, 104101 (2012) PHYSICAL REVIEW LETTERS week ending

7 SEPTEMBER 2012

104101-3

hvi¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

22

c

q;(6)

where cj2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A2

0þB2

0

qjlocates the pinning-depinning

transition. Close to c, the system exhibits a saddle-node

bifurcation with the pattern velocity increasing as the

square root of 2, whereas for larger values of , it behaves

as a linear function of . Relation (6) has been derived

initially in the context of front propagation in patterned

systems [7]. Figure 2shows the excellent ﬁtting (blue solid

line) of hviusing Eq. (6) in the case of Dirichlet boundary

conditions. Using the same ﬁtting, we have a good agree-

ment for the SH model with c¼2:6103(see Fig. 1).

Note that, from Eq. (5) it is also clear that if is equal

to zero, the nonresonant terms vanish. Therefore, the

appearance of the pinning effect in the system is no

longer observed, despite amplitude variations induced by

the boundary conditions. Hence, the phase velocity is line-

arly proportional to , in good agreement with Neumann

conditions.

To provide an experimental veriﬁcation of the pinning-

depinning phenomenon in a drifting monostable pattern

system, we consider the feedback optical system described

in Refs. [26–28] which consists of a Kerr medium sub-

jected to optical feedback provided by a mirror tilted by an

angle [see Fig. 4(a)]. It consists in a nematic liquid

crystal (LC) layer irradiated by a laser beam (F) which is

reﬂected back onto the sample (B) by a simple plane

mirror placed at a variable distance dfrom the LC layer

[Fig. 4(a)]. It is straightforward to derive a similar

amended amplitude equation such as equation (3) from

the model [29] which describes this experimental system.

The full and lengthy expressions of these coefﬁcients, as a

function of the experiment parameters, will be reported

elsewhere [30]. The nonlinear medium is a 50 mthick

layer of E7LC homeotropically anchored with response

time and diffusion length ldwhich are equal to 2:3s

and 10 m, respectively, [31]. The beam is delivered

by a monomode frequency doubled Nd3þ: YVO4laser

(0¼532 nm) which is shaped by means of two cylindri-

cal telescopes in order to achieve a transverse quasi-

monodimensional (1D) pumping (beam diameters

93 m650 m). The optical feedback length dis

equal to 5 mm. The reﬂected beam Bis shifted transversely

with respect to the incoming forward beam F. The trans-

lational shift haccounts for the distance between the two

beams on the LC sample. For a typical feedback length

d¼5cm, the angle is of order of 4 mrad, h2ld¼

20 m(to be compared with the pattern wavelength—

103 m—in the conditions of a uniform pump proﬁle).

In the following, hwill be given in units of ldto keep the

same units as for analytical predictions. The reﬂected beam

is monitored after its second passage through the LC layer

Bout (Fig. 4(a)). The pumping has a Gaussian shape, as a

sort of Dirichlet boundary condition. Associated with the

pattern drift induced by the translational shift h, the system

presents all the ingredients for pinning-depinning transi-

tion. We then focus on the phase velocity evolution of the

convective modes versus the lateral shift h.

As we can see in Fig 4, there is a good agreement

between the predictions and the experimental observations

on all the points. More speciﬁcally, the spatial dependence

of the convective systems leads to the pinning phenome-

non. The measured phase velocity evolves as ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

h2h2

c

p

close to the pinning frontier h2:8ld. This conﬁrms the

observations reported in [14]. In addition, Eq. (6) accu-

rately describes the pinning-depinning transition. For this

particular case, the spatial coupling between the pattern

and the envelope variations is produced by the Gaussian

proﬁle of the pump beam (Fig. 4(b)). In fact, the coupling is

present in almost all the space, given the small number of

pattern wavelengths close to the maximum of the proﬁle of

the beam (aspect ratio 6). Figure 4(c) displays a spatio-

temporal recording of the optical pattern proﬁle exhibiting

a pinned behavior. Above the pinning-depinning transition,

we observe the expected pattern propagation with almost

periodic leaps [Fig. 4(d)]. The aperiodicity comes from

internal noise.

In conclusion, our work generalizes the pinning theory

developed in multistable systems to all systems: Pinning

phenomenon comes from the coupling between the slow

FIG. 4 (color online). Experimental phase velocity of a Kerr

media subject to optical feedback (black dots) and its respec-

tively theoretical ﬁtting [Eq. (6)] (a) Experimental setup. Liquid

crystal (LC) layer; Mfeedback mirror; Finput optical ﬁeld; B

backward optical ﬁeld; Bout output optical ﬁeld sent to CCD

cameras; mirror tilt angle; dfeedback length. (b) Laser inten-

sity proﬁle. Evolution of the experimental phase velocity

versus the translational shift h. The considered parameters

are d¼5mm,I¼634 W=cm2, (c) h¼20:9mand

(d) h¼27 m. Spatiotemporal picture sizes (c), (d) are

590 m800 s.

PRL 109, 104101 (2012) PHYSICAL REVIEW LETTERS week ending

7 SEPTEMBER 2012

104101-4

scale of the pattern envelope to the fast scale of its

modulation.

The authors acknowledge ﬁnancial support by the ANR-

CONICYT 39, ‘‘Colors.’’ M. G. C. and M. A. G-N. are

thankful for the ﬁnancial support of FONDECYT

Projects No. 1120320 and No. 3110024, respectively.

This research was supported in part by the Centre de la

Recherche Scientiﬁque (CNRS) and by the ‘‘Conseil

Re

´gional Nord—Pas de Calais,’’ ‘‘The Fonds Europe

´en

de de

´veloppment Economique de re

´gions.’’C. F. O.

acknowledges the ﬁnancial support of CONICYT by

Beca Magister Nacional and Program of Ayuda de

Estadı

´as Cortas de Investigacio

´n of University of Chile.

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PRL 109, 104101 (2012) PHYSICAL REVIEW LETTERS week ending

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