Localized states in bistable pattern forming systems

0.16 MB·English

by  Umberto Bortolozzo

Localized states in bistable pattern forming systems
U. Bortolozzo1,2, M. G. Clerc3, C. Falcon3, S. Residori1, and R. Rojas1
1Institut Non Lin´eaire de Nice, UMR 6618 CNRS-UNSA,
1361Route des Lucioles, F-06560 Valbonne, France.
2Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6 50125 Florence, Italy
3Departamento de F´isica, Facultad de Ciencias F´isicas y Matem´aticas,
Universidad de Chile, Casilla 487-3, Santiago, Chile.
We present an unifyingdescription close to a spatial bifurcation of localized states, appearing as
largeamplitudepeaksnucleatingoverapatternofloweramplitude. Localizedstatesarepinnedover
alatticespontaneouslygeneratedbythesystemitself. Weshowthatthephenomenonisgenericand
requiresonlythecoexistenceoftwospatiallyperiodicstates. Attheonsetofthespatialbifurcation,
6 aforced amplitudeequation is derived for thecritical modes, which accounts for theappearance of
0 localized peaks.
0
2
n During the last years emerging localized structures in existence between different patterns and stable front so-
a dissipativesystemshavebeenobservedindifferentfields, lutionsbetweenthem,isbasedonanamplitudeequation
J suchasdomainsinmagneticmaterials[1],chiralbubbles thatincludesaspatialparametricforcing. Thisextension

0 in liquid crystals [2], current filaments in gas discharge withrespecttoconventionalamplitudeequations,allows
3 experiments [3], spots in chemicalreactions[4], localized todescribelocalizedpatternsandtoaccountforthemain
2D states in fluid surface waves [5], oscillons in granular properties of these solutions. The model includes the in-
]
S media [6], isolated states in thermal convection [7], soli- teraction of the slowly varying envelope with the small
P tary wavesin nonlinear optics [8], just to mention a few. scale of the underlying pattern solution [16], well-known
. Inone-dimensionalsystems,localizedpatternscanbede- as the non-adiabatic effect [17, 18].
n
scribed as homoclinic orbits passing close to a spatially
i
l oscillatorystateandconvergingtoanhomogeneousstate
n
[9, 10], whereas domains are seen as heteroclinic trajec-
[
tories joining the fixed points of the corresponding dy-

2 namicalsystem[11]. Recently,inanematicliquidcrystal
v
lightvalvewithopticalfeedbackithasbeenfoundexper-
4
imentally a different type of localized states, appearing
5
0 as a large amplitude peaks nucleating over a lower am-
1 plitude pattern and therefore called localized peaks [12].
1 Similar observations have been reported in a Newtonian
5 fluid when non linear surface waves are parametrically
0
excited with two frequencies [13] and in numerical sim-
/
n ulations of an atomic vapor with optical feedback [14].
i Recently, longitudinal modes with localized peaks over
l
n a spatially modulated background have been shown in
v: numerical simulations of Maxwell-Bloch equations for a FIG. 1: A typical bifurcation diagram allowing for the ap-
i semiconductor laser [15]. pearance of localized peaks: at a certain value of µ a sec-
X Allthesedifferenttypesoflocalizedstatesappearover ondary subcritical bifurcation takesplace; dashed lines mark
r a patterned background and thus constitute a different the beginning (end) B1 (B2) of the bistable region and the
a classofstructureswithrespecttotheonesappearingover Maxwell point µM.
anuniformbackground. Theaimofthismanuscriptisto
Generally, the main ingredient for the appearance of
show that localized peaks are a generic class of localized
localized peaks is the coexistence between two spatially
states,appearingwheneverapatternformingsystemex-
periodic states. In order to give a generic description of
hibits coexistence of two spatially periodic states. The
such a situation, we consider a system that exhibits a
mechanisms that originate this circumstances are more
sequence of spatial bifurcations as shown in Fig.1, that
than a few, for instance, one can consider a multi-stable
is,the primarybifurcationissuper-criticalwhile the sec-
system,whichshowstwoconsecutivespatialbifurcations
ondary one is of subcritical type. Let ~u(x;t) be a vector
todifferentstateswhenoneparameterischanged. There
field that describes the system under study and satisfies
is a large number of physical systems that display this
the partial differential equation
kindofbehavior,thereforethereisavastnumberofpos-
sible models. In order to derive an unifying and sim- ∂ ~u=f~(~u,∂ , λ ), (1)
t x i
ple description of localized peaks, we develop a theoreti- { }
calmodelforone-dimensionalspatiallyextendedsystems where λ is a set of parameters. For a critical value
i
{ }
closetoaspatialbifurcation. Themodel,whichshowsco- of one of the parameters, the system exhibits a spatial
2
instability at a given wave number q . Close to this spa-
c
tial instability, we use the ansatz ~u = A(X,T)eiqcxuˆ+
A¯(X,T)e−iqcxuˆ¯+ and the amplitude satisfies [9]
···
∂ A=µA ν A2A+αA4A A6A+∂ A, (2)
T XX
− | | | | −| |
where µ is the bifurcation parameter and ν,α con-
{ }
trol the type of the bifurcation (first or second order de-
pending on the sign of these coefficients). Higher-order
terms are ruled out by scaling analysis, since ν µ2/3,
α µ1/3, A µ1/6, ∂ µ, ∂ µ1/2, and ∼µ 1.
t x
No∼te that t|hi|s∼approach is∼phase in∼variant (A A≪eiϕ),
→
but the initial system under study does not necessarily
FIG. 2: Localized state and pattern solutions: a) localized
have this symmetry.
state solution between pattern state Π1 and Π2; b) and c)
As depicted in Fig.1, for a given range of parameter
represent the pattern solutions Π1 and Π2 coexisting for the
values the system exhibits coexistence between two dif-
same parameters.
ferent spatially periodic states, each one corresponding
to a homogeneousstate for the amplitude equation. The
coexistence region is for B1 < µ < B2. The extended where ∆ is the distance between the cores of each front,
stationarysolutionoftheamplitudeequationEq.(2),has δ is the separation from the Maxwell point (µ µ ), λ
M
−
the form (∂tA=0) characterizestheexponentialdecayofthefronttoagiven
constant value at infinity, and a is a positive coefficient
i ε X
A=Roe R2o , that characterizes the properties of the interaction and
is determined by the form of the front. The Eq.(3) has
where µ ε2/R4 νR2+αR4 R6 =0 andε is anarbi-
− o− o o− o an unstable fixed point ∆∗ = ln(δ/a)/λ, which is the
trary constant related to the initial phase invariance. It −
nucleation barrier between the two homogeneous states.
is worth to note that in the case of positive ε, the wave
Hence,theconventionalamplitudeequation,Eq.(2),does
number of the pattern is modified by the inverse of the
not exhibit stable localized states, due to the scale sepa-
squareamplitudeR2,sothatpatternswithlargerampli-
0 ration used to derive the amplitude equation. But near
tude have smaller wave number. At variance when ε is
the front’s core, the previous ansatz is no more valid.
negative,the patterns with largeamplitude havesmaller
Indeed, in these locations the slowly varying envelope
wavelength. In Fig.2 are depicted two different patterns
A(X,T) shows oscillations of the same (or comparable)
that coexist for the same parameters and the pattern
size as the small scale of the underlying pattern. This
with large amplitude has smaller wavelength, hence ε in
phenomenon is denominated as the non-adiabatic effect
this case is negative.
[17, 18].
Note that the above amplitude equation is variational
Inordertotakeintoaccountthiseffect,wemodifythe
and can be written as
amplitudeequationbyincludingthenon-resonantterms.
δ A, A Thus, the amplitude equation becomes
∂ A= F ,
t − (cid:2)δ A (cid:3) ∂ A = µA ν A2A+αA4A A6A+∂ A
T XX
− | | | | −| |
where + gmnAmA¯ne−iqc(1+√nµ−m)X (4)
= µA2 ν|A|4 +α|A|6 |A|8 ∂ A 2 dx. mX,n≥0
X
F −Z (cid:18) | | − 2 3 − 4 −| | (cid:19)
where g are real numbers of order one. Now the
mn
Forgivenvaluesofthe parameters,thetwostableuni- amplitude equation is parametrically forced in space
form stationary states of Eq.(2) have the same energy, by the non-resonant terms. We note that the ansatz
that is, the system is at the Maxwell point. Where the for ~u satisfies the symmetries x x, A A¯ , and
→− →
frontbetweenthe two statesdoes notpropagate,thatis, x x+xo, A Aeiqcxo . T(cid:8)herefore, the en(cid:9)velope
→ →
the front is motionless [19]. By moving away from the e(cid:8)quation also is invariant u(cid:9)nder this transformation. In-
Maxwell point, the front dynamics is usually character- stead, the spatial translation and phase invariance are
ized by the motion of the core of the front, which is de- independent symmetries of Eq. (2).
finedasthefrontpositionwiththelargestslope. Inorder Tounderstandandillustratetheeffectofnon-resonant
to have a localized solution, we consider the interaction terms we keep the leading term n=0 and m=2. Then
of two of these motionless fronts close to the Maxwell the amplitude equation takes the form
point. As a consequence of the asymptotic behavior of
∂ A = µA ν A2A+αA4A A6A+∂ A
the front at infinity, the front interaction is attractive, T − | | | | −| | XX
and has the form [20] +ηA2ei√qcµX. (5)
∆˙ = ae λ∆+δ, (3) The amplitude is now spatially forced with frequency
−
−
3
with
2 3µR+2 5νR+4 +7αR+6 3εR+−4 ∂xR+
a = − h − − | i,
∂xR+ ∂xR+
h | i
F(R+) F(R )
δ = − − ,
∂xR+ ∂xR+
h | i
γ = ηh∂xR+|R+2 cos(cid:16)√qcµx(cid:17)i,
∂xR+ ∂xR+
h | i
F(R)=µR2/2 νR4/4+αR6/6 R8/8+2ε2/R2, and
− −
f g ∞ f(x)g(x)dx.
h | i≡
As a cRo−n∞sequence of the spatialforcing the interaction
of two fronts close to the pinning range, Eq.(6), has an
extra term and now alternates between attractive and
repulsiveforces. Itisimportanttoremarkthatγ isapa-
rameter exponentially small, proportionalto η, and is of
orderδ,i.e. thesourceoftheperiodicalforceisthespatial
FIG. 3: Oscillatory interaction force between two front so-
lutions. The inset figures are the stable localized patterns forcing in the Eq.( 5). Therefore, close to the Maxwell
observedattheMaxwellpoints(blackdots),wheretheinter- point the system exhibits a family of equilibrium points,
action changes its sign. d∆/dt=0. Eachequilibriumpointcorrespondtoalocal-
izedsolutionnucleatingoverapatternstate,wecallthese
solutionslocalized patterns. The lengths oflocalizedpat-
qc/2π√µ and amplitude η g02. The spatial forcing ternsaremultiple ofabasiclength,correspondingtothe
≡
isresponsibleforthehomogenousstatesbecomingaspa- shortestlocalizedstate. Wetermtheseshorteststatesas
tiallyperiodicstate. Asaconsequence,thefrontsolution localized-peaks, as these solutions correspond to the ex-
between the spatially periodic states exhibits a pinning perimental observations reported in [12]. In Fig.3, it is
range, that is, the front is motionless for a range of pa- depicted the front interaction and the family of equilib-
rameter around the Maxwell point. It is important to rium points.
remark that the model (5) is the simplest model that Due to the oscillatory nature of the front interaction,
exhibits front solution between two different spatial pe- which alternates between attractive and repulsive forces
riodic solutions. (cf. Fig.3), we can deduce the dynamical evolution and
Note that the maxima of the envelope correspond to bifurcation diagram of localized patterns. By decreasing
the maxima of the initial periodic solution ~u(t,x). In δ or increasing η, the family of localized patterns disap-
order to obtain the change of the front interaction as a pearsbysuccessivesaddle-nodebifurcationsandonlylo-
resultofthespatialforcing,weconsiderthefrontsolution calizedpeaks survive. The mechanismfor localizedpeak
of the resonant equation appearance is related to the fact that the spatial forcing
satAis±fi(exs−xo)=R±(x−xo)eiR ε/R2±dx,whereR±(x−xo) itshenopnaltitneeranr.amInpdlieteudd,ei.t is proportional to the square of
ε2 Since the amplitude of spatial forcing for the upper
µR−νR3+αR5−R7+∂xxR− R3 =0, branchis largerthan that for the lower branch,then the
patternswithlargemodulushavelargespatialamplitude
xo isthe positionofthe frontcoreandthelowerindex + oscillations around the equilibrium state of unperturbed
( ) correspond to a front monotonically rising (decreas- amplitude equation (cf. Fig.(2)). Thus, for a given criti-
−
ing). As the non resonantterm is a rapid spatial oscilla- cal,andsmall, value ofthe forcingthe pattern withhigh
tion, we consider this term as perturbative-type and use magnitude becomes unstable, because this state collides
the anstaz with the unstable pattern state. In Fig. (1), this unsta-
ble state is represented by the dashed line. Hence, the
A=A+(x−x1(t))+A−(x−x2(t))−(Ao,+−Ao,−)+δWeiδϕ, minima of the pattern with high magnitude reach the
ainrethsemEalql.(fu5)n,cwtihonerse, Aanod,±R=Roa,±reeitεhxe/Rs2ot,a±b,laenedqu{δilWib,riδuϕm} pcaattitoenrnowf itthhelospwaetriamlapgenriitouddice,saonludtiaonsagdidveles-nroisdeetboiftuhre-
statesoftheresonantampliot,u±deequation(2)andRo,+ > anpispme,arloacnacleizoefdapalottcearlnizsewdiptheaaks.izBeelcaarguesretohfatnhtishemsehcohrat--
R . We obtain the following solvability condition for
thoe,−δW function (front interaction) estlengtharenotrobustphenomena. Infact,thetypical
behavior observed in the experiments is the appearance
of localized-peaks [12].
∆˙ = ae λ∆+δ+γcos qc ∆ , (6) In Fig.4a, it is shown a localized peak profile recorded
−
− (cid:18)√µ (cid:19) in the Liquid-Crystal-Light-Valve (LCLV) experiment
4
[12]. In order to directly compare with the model, we solutionthatconnectsthetwodifferentpatternstatesex-
have performed one-dimensional experiments by insert- hibit a locking phenomena, that is, it is motionless for a
ing a rectangular slit in the optical feedback loop. The range of parameter. We have obtained the front inter-
slit transversesize is approximately100µm whereas,for action and from this interaction we have deduced the
the parameters set in the experiment, the size of local- family of localized solutions. We have shown that, as a
ized peaks is around 350 µm. A similar profile can be consequenceofthe nonlinearnatureofthe forcing,local-
numericallyobtainedfor ~u 2 = A 2 cos(q x), asshown ized patterns with a size larger than the shortest length
c
k k | |
in Fig.4b. are not robust phenomena, so that only localized peaks
are stable at long times and for a wide range of parame-
ters. We have shown a good qualitative agreement with
the experimental observations for a LCLV system and
we expect similar phenomena to be observed in other
pattern forming systems, provided they present bistabil-
ity between two different spatial structures. Note that
pinning of localized structures on periodic arrayshas re-
centlybeenreportedforafixedgrid[21]. Localizedpeaks
can be seen as a generalization of this case, when the
pinning lattice is spontaneouslygeneratedby the system
FIG. 4: a) Intensity profile of a one-dimensional localized
peak in the LCLV experiment; b) k~uk2 numerical profile in itself.
thepresence of a localized peak.
The simulation software DimX, developed at INLN,
In conclusion, we have presented an unifying descrip- has been used for all the numerical simulations pre-
tion of localized peaks, which are large amplitude peaks sented in this paper. The authors thanks the sup-
nucleating over a lower amplitude pattern. We have de- port of ECOS-CONICYT collaboration program. M.G.
rived a spatially forced amplitude equation and shown C. acknowledges the financial support of FONDECYT
that localized peaks are a generic class of behavior ap- project 1051117, and FONDAP grant 11980002. R.R.
pearing whenever a pattern forming system exhibits co- acknowledges financial support from Beca Presidente de
existencebetweentwospatiallyperiodicstates. Thefront la Repu´blica of the Chilean Government.
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