on homoclinic snaking in optical systems

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On homoclinic snaking in optical systems W. J. Firth, L. Columbo, and T. Maggipinto Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 17, 037115 (2007); doi: 10.1063/1.2768157 View online: http://dx.doi.org/10.1063/1.2768157 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/17/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes Chaos 17, 037112 (2007); 10.1063/1.2746830 Homoclinic snaking: Structure and stability Chaos 17, 037102 (2007); 10.1063/1.2746816 Optical bistability and stationary patterns in photonic-crystal vertical-cavity surface-emitting lasers Appl. Phys. Lett. 86, 021116 (2005); 10.1063/1.1853509 On thermal convection in slowly rotating systems Chaos 14, 803 (2004); 10.1063/1.1774413 Localized structures in nonlinear optics: spatial features and interactions AIP Conf. Proc. 622, 299 (2002); 10.1063/1.1487546 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 23:15:02

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Page 1: On homoclinic snaking in optical systems

On homoclinic snaking in optical systemsW. J. Firth, L. Columbo, and T. Maggipinto Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 17, 037115 (2007); doi: 10.1063/1.2768157 View online: http://dx.doi.org/10.1063/1.2768157 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/17/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsatingcomplexes Chaos 17, 037112 (2007); 10.1063/1.2746830 Homoclinic snaking: Structure and stability Chaos 17, 037102 (2007); 10.1063/1.2746816 Optical bistability and stationary patterns in photonic-crystal vertical-cavity surface-emitting lasers Appl. Phys. Lett. 86, 021116 (2005); 10.1063/1.1853509 On thermal convection in slowly rotating systems Chaos 14, 803 (2004); 10.1063/1.1774413 Localized structures in nonlinear optics: spatial features and interactions AIP Conf. Proc. 622, 299 (2002); 10.1063/1.1487546

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On homoclinic snaking in optical systemsW. J. FirthSUPA and Department of Physics, University of Strathclyde, 107 Rottenrow, G4 0NG Glasgow, Scotland

L. Columbo and T. MaggipintoCNR-INFM LIT3, Dipartimento di Fisica Interateneo, Università degli Studi di Bari, via Amendola 173,70126 Bari, Italy

�Received 9 March 2007; accepted 11 July 2007; published online 28 September 2007�

The existence of localized structures, including so-called cavity solitons, in driven optical systemsis discussed. In theory, they should exist only below the threshold of a subcritical modulationalinstability, but in experiment they often appear spontaneously on parameter variation. The additionof a nonlocal nonlinearity may resolve this discrepancy by tilting the “snaking” bifurcation diagramcharacteristic of such problems. © 2007 American Institute of Physics. �DOI: 10.1063/1.2768157�

In this paper, we are mainly interested in localized statesin systems where subcritical pattern formation is ex-pected. In experiments, the abrupt appearance of a fullydeveloped pattern is not always observed. Instead, as thedriving parameter is increased, localized states may ap-pear spontaneously, and then multiply as the system isdriven harder, only eventually merging into an extendedpattern, if at all. In contrast, corresponding models usu-ally exhibit such localized states only in the region whereboth patterned and unpatterned states are stable, so thatthe localized states should only be accessible by hard ex-citation (addressing). We discuss this discrepancy, and itspossible origins, primarily in relation to optical systems.We propose that it may be resolved by the addition of aninhibitory nonlocal (quasiglobal) coupling term to the ex-isting models, and present some preliminary results sup-porting this hypothesis.

I. INTRODUCTION

Pattern formation is a widespread consequence of non-linearity in spatially extended systems,1 and has been pre-dicted and/or observed in a wide variety of systems frommany fields, including fluid dynamics,2 chemistry,3 biology,4

ferrofluids,5 gas discharges,6 and optics.7–20 When a systemparameter is varied in simulations of model systems, typi-cally a pattern appears spontaneously at the modulationalinstability �MI� threshold. The pattern may grow smoothlyfrom small amplitude, but sometimes there is an abruptswitch into a large-amplitude pattern. This persists as theparameter is reduced back below the switching threshold,until an abrupt collapse to the unpatterned state at a saddle-node �SN� bifurcation. In such subcritical cases,1 both pat-terned and unpatterned states are stable over a finite param-eter range, between SN and MI. In this paper, we will beprimarily concerned with nonlinear optical systems. For atypical optical model �saturable absorber in acavity13,15,16,19�, state diagrams for subcritical stripe andhexagon output patterns are shown in Fig. 1 as a function ofdrive intensity parameter I and relative wave vector K /Kc,

where Kc is the critical wave vector, i.e., that with the lowestMI threshold.

In this paper, we are mainly interested in localized states�LS� in systems where subcritical pattern formation is ex-pected. In experiments, the abrupt appearance of a fully de-veloped pattern is not always observed. Instead, as the driv-ing parameter is increased, LS may appear spontaneously,and multiply as the system is driven harder, perhaps eventu-ally merging into an extended pattern. Figure 2 is an examplein a gas-discharge system,21 Figs. 3 and 4 from optics.22,23

We conjecture that this discrepancy between theory andexperiment is due to an additional nonlocal �or quasiglobal�nonlinear coupling that favors LS over system-wide patterns.We develop that idea, and demonstrate its potential applica-bility, in later sections of the paper.

The paper is organized as follows. In Sec. II, we dicussthe phenomenon of homoclinic snaking, which is the typicalscenario for the occurrence of sequences of LS in systemsshowing subcritical pattern formation. In Sec. III, we presentand describe several basic optical models in which ho-moclinic snaking is found. In Sec. IV, we show that aug-menting these models with a global coupling can tilt thehomoclinic snakes, with the result that LS, rather than a pat-tern, appear as the system is driven beyond MI threshold.Finally, in Sec. V we discuss the effects of finite-range non-locality, mediated by a symmetric kernel. We also considersome physical mechanisms for the sort of nonlocal or quasi-global nonlinearity that could account for the spontaneousappearance of LS in experiments.

II. HOMOCLINIC SNAKING

Localized states in models where a pattern is in compe-tition with a homogeneous state have been the subject ofintense recent interest.14–16,18,24–31 Coullet et al.25 proved theappearance, between SN and MI, of a multiplicity of LS,which resemble subsections of the pattern. This work builton Pomeau’s realization32 that the interface between a patternand a flat state can be stationary over a finite parameterrange, the “locking range.” Similar scenarios were previ-ously identified for localized buckling26 and self-replicating

CHAOS 17, 037115 �2007�

1054-1500/2007/17�3�/037115/8/$23.00 © 2007 American Institute of Physics17, 037115-1

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patterns.27 In one spatial dimension �1D�, there are typicallytwo sequences of LS, with, respectively, even and odd num-bers of peaks. We will use LSn to denote an LS with n mainpeaks.

Figure 5 demonstrates this phenomenon for the saturableabsorber model of Fig. 1. Within each sequence, the energy�or other norm� characteristically “snakes” upwards, zigzag-ging to and fro across the locking range, adding a pair ofpeaks on each positive-slope “zig,” while the connectingnegative-slope “zags” are always unstable.

Because all of these LS are homoclinic to the flat state,the phenomenon illustrated in Fig. 5 is often termed ho-moclinic snaking. �See Ref. 30 for a comprehensive accountof homoclinic snaking in a Swift-Hohenberg model.�

FIG. 1. Existence diagrams for �a� stripes and �b� hexagons for a saturableabsorber in an optical cavity, as a function of drive intensity I and relativewave vector K /Kc, where Kc is the critical wave vector �Ref. 19�. In eachcase, the MI threshold line is labeled. The pattern is subcritical for I betweenits minimum and the saddle-node �SN� lines �lowest curves�. Shadings in-dicate regions where the pattern is unstable �Ref. 19�. hexagons are stableover a broad band of wave vectors; stripes are less stable, and the hexagonis unstable below the Irh curve.

FIG. 2. Diagram of intensity current against applied voltage in a gas dis-charge system showing a bifurcation sequence of current filaments �courtesyof H. -G. Purwins �Ref. 21��.

FIG. 3. Inverted contrast images showing a sequential appearance of local-ized states �intensity spots on a flat background� in an optically pumpedsemiconductor laser amplifier as the pumping rate is increased, leadingeventually to a pattern-like state �Ref. 22�.

037115-2 Firth, Columbo, and Maggipinto Chaos 17, 037115 �2007�

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As the number n of peaks increases, the central part ofeach LSn resembles more and more a coexistent roll pattern,while at each wing, it approaches asymptotically the station-ary front between the patterned and the unpatterned statesthat characterizes the locking range.25,30 In this regime, thesnaking can be quantified by delicate beyond-all-ordersasymptotic theory.31 Our main interest here, however, is inthe few-peak LS forming the lower portions of the snakes. Inthe model systems to which the above theory25 applies, theirexistence range is smaller than, and lies wholly within, therange �SN,MI� over which both patterned and unpatternedstates are stable, see, for example, Figs. 1 and 5. To observesuch LS, it should therefore be necessary to place the controlparameter within the snaking range, and apply a local exci-tation in the form of an address pulse. Under appropriateconditions, the system will evolve during and after the ad-dress pulse in such a way as to end on the desired “zig” ofthe snake. �To generate the LS shown in Fig. 5, the systemwas initialized with an LS-like structure, and a Newtonmethod used to converge to the nearest LS,28 which re-sembles somewhat the address-pulse procedure.� LS appear-ance in experiments as a parameter is changed, without ad-dressing, is thus unexpected. It might lead one to questionthe applicability of homoclinic snaking to experiment.

It is necessary to remark that the theory of homoclinicsnaking is strictly applicable only in 1D, because it is builton the powerful methods of reversible dynamical systemstheory, using an analogy between time and 1D space. Nev-ertheless, localized states sitting on a flat background canonly be stable if that background is stable, regardless of spa-tial dimensionality. Equally, a locking range between a flatsolution and a coexistent pattern �e.g., hexagons� can also beexpected in 2D if the pattern is subcritical. It is therefore nosurprise that 2D LS are found in such model systems, inassociation with a locking range, and existing only well be-low MI. Figure 6 shows the bifurcation structure of 2D LSwith a single peak �LS1�, and linear, triangular, and rhombicclusters that look like bound states of LS1 units.

The data of Fig. 6 are again for the optical cavity withthe saturable absorber model, where it was also shown19 thathexagons invade the flat state for I within about 15% of theMI threshold. Below this I, a “cracked” hexagon pattern, i.e.,islands of hexagons with flat-state between, was found to bestable, clearly illustrating stable locked fronts �see Fig. 7�.

Thus it seems that LS, and in particular the basic LS1,should exist only below MI, and should appear only by lo-calized addressing, whether in 1D or 2D. LS1 have beencreated �and extinguished� by local perturbations in opticalsystems, e.g., Refs. 33 and 34, and elsewhere, e.g., in a

FIG. 4. Sequential appearance of lo-calized states with increasing currentin an electrically pumped semicon-ductor laser amplifier �Ref. 23�. �Theinput beam phase is structured to trapLS on an array.�

FIG. 5. Integral norm of 1D localized structures against intracavity back-ground intensity �from McSloy et al. �Ref. 28��.

FIG. 6. Bifurcation diagram of 2D LS. Inset: examples of 2D LS clusterscorresponding to the positive stable branches �from McSloy et al. �Ref. 28��.

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ferrofluid.5 Thus LS1 exist as subcritical structures in at leastsome cases. The spontaneous appearance of LS1-like objectson parameter variation, as in Figs. 2–4, is, however, incon-sistent with the bifurcation structure found in 2D modelssuch as that of Fig. 6.

An obvious explanation for this discrepancy betweentheory and experiment is experimental imperfection. Perhapsthe MI threshold varies across the system in such a way thatpatterns can form only at local “sweet spots.” This is ofcourse possible, but experimentalists are well aware of suchpitfalls, and take great care to minimize inhomogeneities.Furthermore, the subcritical pattern should not be confined tojust those areas where the MI threshold is locally exceeded.It should invade the surrounding region, stopping only at the“locking point,” where the front between it and the unpat-terned state is stationary. For this expansion to halt whenonly a single spot has formed, as in Figs. 2–4, would implyinhomogeneities so strong that the basic concepts of patternformation would be barely applicable to these experiments.Further, the LS that appear spontaneously at different loca-tions seem remarkably similar to one another, which wouldbe hard to explain if they were imperfection-generated.

Less obvious, but perhaps a more likely explanation, isthat there is some intrinsic feature of the experiments notincluded in the basic models. Suppose, for example, that thepresence of one LS inhibits the formation of a second in itsneighborhood. There would be no effect on the MI threshold,but the development of a pattern would be inhibited oversome effective range. Isolated LS1 would form, coalescing toform a pattern only if and when the MI dynamics becomesstrong enough to overcome the inhibition mechanism. Sucheffects, which could obviously account for experimentaldata, are found below when a long-range �quasiglobal� cou-pling is added to an optical model.

III. OPTICAL MODELS EXHIBITINGHOMOCLINIC SNAKING

In nonlinear optical cavities, the dynamical variable isusually a complex field amplitude E. We will consider onlymodels in which the optical frequency is defined by a mono-chromatic input field of amplitude EI, which thus excludesconsideration of LS in lasers �see, e.g., Ref. 35�. We will alsolimit ourselves to examples in which the longitudinal depen-dence of E may be eliminated by a suitable averaging �so-called mean-field models�, so that for us space will be 2D�x ,y� at most, or 1D �x� if the y dependence is trivial orstrongly constrained.

Nonlinearity may involve a dependence on E of eitherthe absorption or the refractive index of a medium within thecavity, e.g., the Kerr effect changes the refractive index inproportion to �E�2. Such a nonlinearity is necessarily local,the optical properties at any point depending only on thefield at that point. In many cases, the nonlinearity is medi-ated by an excitation, such as when E interacts with electronsand holes in semiconductor media. If such an excitation ismobile, it may diffuse, or be driven, transversely, and mayalso have dynamics, and so models described by coupledfield-population equations are also important in optical sys-tems.

We first consider the saturable absorber model alreadyfrequently instanced in this paper.13,15,16,19,28 It directly de-rives from the mean-field limit of the Maxwell-Blochequations36 after adiabatic elimination of the atomic vari-ables, for the special case in which the atoms are driven attheir resonance frequency, and so their response is purelyabsorptive:

�E

�t= − E��1 + i�� +

2C

1 + �E�2� + EI + ia�2E + � . �1�

Here time is scaled to the decay time of the empty cavity; �describes the detuning between the input field and the nearestcavity resonance; 2C parametrizes the density of the intrac-avity saturable absorber, which bleaches when �E� gets large,hence making �1� nonlinear. �E is scaled such that the ab-sorption is 50% of its small-signal level when �E�=1.� TheLaplacian term describes transverse coupling due to diffrac-tion: we will choose spatial scaling such that a=1. The finalterm � provides for a global coupling term not present inprevious work, to be introduced in due course below. We willassume EI constant and spatially uniform, so that there is acorrespondingly uniform intracavity field E0. For appropriateparameters, E0 is unique, and then I= �E0�2 is a convenientsweeping parameter �and has been used as such in several ofthe above diagrams�.

Meanwhile, setting �=0, system �1� for negative �shows MI to patterns with wave vector obeying Kc

2= �−��with threshold IMI, the smaller root of �I+1�2=2C�I−1�. Wetake C=5.4, giving IMI�1.65, and �=−1.2 �for which I is asingle-valued function of EI�. Figure 5 shows homoclinicsnaking for this model in 1D, in terms of the integral modu-lus of the zero-background auxiliary field A�x�, where E�x�=E0�1+A�x��. Importantly, this model also exhibits patterns�Fig. 1 and 7� and stable LS.15,16 In 2D, a phenomenon re-

FIG. 7. “Cracked” hexagon pattern showing the existence of large-scalelocked fronts between the patterned configuration and the flat background�Ref. 19�.

037115-4 Firth, Columbo, and Maggipinto Chaos 17, 037115 �2007�

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sembling homoclinic snaking also occurs, as noted a decadeago for cylindrically symmetric LS15 and for cluster states inRef. 28 and illustrated in Fig. 6. Note that in optics, thesingle bright spot LS1 is usually called a cavity soliton �CS�.

As a second optical model, we consider a cavity contain-ing a Kerr medium. The field amplitude in an extended Kerrmedium is described by the nonlinear Schrödinger equation�NLS�, and thus supports stable solitons in 1D, withself-focusing collapse in 2D for a positive nonlinearity.Cavity dissipation stops collapse, and stable CS are found inboth 1D and 2D models.7,29 The field equation for a Kerrcavity is8

�E

�t= − E��1 + i�� − i�E�2� + EI + ia�2E , �2�

which differs from �1� only in the nonlinear term. It is atwo-parameter model, with I and � as convenient parameters.I is unique for ��3, and the MI threshold is I=1. The 1Droll pattern is subcritical for ��41/30. In Fig. 8, we repro-duce the projections onto the �I ,�� plane of the SN linescorresponding to low-order LSn.37 Since these SN lines arethe signature of snaking, this diagram is a reminder that ho-moclinic snakes are in fact one-dimensional sections of asurface in a parameter space that is multidimensional in gen-eral �here, for the Kerr cavity, two�.

The experimental LS in Figs. 3 and 4 and in Ref. 33were obtained in semiconductor microresonators, importantfrom a practical point of view since they allow for miniatur-ization and fast response, and their growth can be controlledwith a high level of accuracy. In a key optical experiment,independent writing of two separate CS and their subsequenterasure by reversing the phase of the writing field wasdemonstrated33 in a setup similar to that of Fig. 4. As a third

optical model, we therefore consider a broad-area vertical-cavity semiconductor microresonator of the Fabry-Perottype, driven by an external coherent field and containing abulk layer of GaAs as the active medium. Adopting again themean-field limit, the dynamical equation governing the elec-tric field inside the cavity and the carrier density of the activematerial are38

�E

�t= − �1 + i��E + EI + i��nlE + i�2E , �3�

�N

�t= − ��N − Im��nl��E�2 − d�2N� , �4�

where � is termed the bistability parameter; N is the carrierdensity scaled to its transparency value; � is the carrier decayrate scaled on the cavity decay rate and d is the carrier dif-fusion coefficient. The nonlinear response of the bulk mate-rial is described by the complex susceptibility �nl�N ,��, �being a normalized band-gap detuning. The susceptibility ismodeled in the framework of the free-carrier and quasiequi-librium approximation and adopting the Urbach tail and theband-gap renormalization.38 For d=0 and steady state, thecarrier equation can be solved for N��E�2� and substitutedinto the field equation, which then becomes somewhat like acombination of saturable absorber and Kerr models, since ithas a local but complex nonlinearity.

This model has been extensively analyzed both in 1Dand 2D;39–41 it exhibits plane-wave instability and MI for awide range of parameter choices and for injection frequen-cies both above ���0� and below ���0� the band gap. Inparticular, we consider here the following set of parameters:�=1, �=80, �=−9, d=0.2, and �=0.0014, which have beenproven to fit experimental conditions quite well.38,39 Thesteady-state curve of the homogeneous solution is bistable,and there is a branch of stable LS1 asymptotic to the lowerstable homogeneous solution, and associated with a stablesubcritical pattern solution.

There has been no previous study of multipeak LS forthis model. Investigating such states in 1D, we find typicalLS homoclinic snaking, which is shown in Fig. 9, where wedisplay the LS existence branches by plotting, againstthe intensity parameter I, the excess over I of average inten-sity Ia.

IV. “SIDEWINDER” SNAKINGIN A SIMPLE OPTICAL MODEL

In contrast with the above models, Figs. 3 and 4 demon-strate that CS can appear spontaneously, without addressing,on parameter variation. In these experiments, snaking is ei-ther absent or the snake is somehow “tilted,” so as to over-hang the MI threshold. Such a snake might be termed a“sidewinder.”

Looking for a theoretical basis for the above behavior,we introduce in the model equation �1� a nonlocal couplingterm � having the following form:

FIG. 8. Projection in the �I ,�� plane of the SN for low-order LSn found inthe Kerr cavity �from Gomila et al. �Ref. 37��. The insets give a magnifiedview of the sequence of saddle-node bifurcations delimiting the homoclinicsnaking.

037115-5 On homoclinic snaking in optical systems Chaos 17, 037115 �2007�

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��x,y,t� = i1

S�

S

E�x�,y�,t�E*�x�,y�,t�K�x,y,x�,y��dx�dy��E�x,y,t� = iIaE�x,y,t� , �5�

where S is the measure of the spatial integration domain and is a real constant coefficient. Since we suppose here thatthe kernel K is a real constant equal to 1, the term � is pureimaginary and represents a global, nonlinear term whose ac-tion is to effectively change the cavity detuning �. For thecase of interest here, ��0, � further detunes the cavity if�0, hence raising the MI threshold for a given inputfield EI.

Global coupling is, of course, physically unrealistic, butshould provide a good guide to the effects of a nonlocalcoupling with a range larger than the size of the LS consid-ered.

Setting equal to zero the time derivatives in Eq. �1�yields the condition for stationary solutions. We study theirexistence, and their stability with respect to spatially modu-lated perturbations, by means of a Newton method;42 in Fig.10, we report the results obtained in 1D for =0.25, �=−1.0, C=5.4 together with the LS stable and unstable ex-istence branches. We plot the difference Ia− I against I, as-suming that the contribution to � arising from I is incorpo-rated into �. Analogously to Figs. 5 and 9, LS still form twointertwined snakes, bifurcating from the point of modula-tional instability �IMI=1.66 for parameters in Fig. 10� andassociated with even or odd numbers of intensity peaks. Theglobal coupling has, however, tilted the snakes, to form asidewinder that overhangs the MI threshold.

We confirm this behavior by numerical integration of thefull partial differential equation �1� with a global couplingterm given by �5�. Adding noise to a stationary stable stateon the homogeneous branch brings the system on to the LS1

state, then increasing EI induces a switching sequence thatprogressively adds single peaks to the previous configuration�see Fig. 11�.

Following the “sidewinder” upwards, we reach the rollpattern branch, where it ends. As expected on the basis of theprevious considerations, and contrary to what happens for�=0, the roll pattern is stable only well beyond the MI point.

Turning to the stability of the sidewinder states, we re-port in Fig. 12 the eigenvalues with the largest real parts forthe first few folds of the snake corresponding to an odd num-ber of peaks. These are found from the Jacobian matrix of aspatial discretized version of Eq. �1�. Apart from the constantpresence of a null eigenvalue linked to the translational sym-metry of the problem, we see a scenario richer than thatdescribed, for example, in Ref. 28. The other two eigenval-ues plotted correspond to “even” �symmetry-preserving� and“odd” or �symmetry-breaking� Eigenmodes. Only the evenmode has positive growth rate; the LS destabilizes throughthe growth or decay of a pair of peaks at its margin. The oddmode, however, changes the peak number by one unit only,

FIG. 9. Homoclinic snaking of 1D LS in a bulk semiconductor microreso-nator. Against the intensity parameter I is plotted the excess spatially aver-aged intensity Ia− I for the various LS configurations. Parameters: �=1, �=80, �=−9, d=0.2, and �=0.0014. The uppermost curve represents a part ofthe stable branch of a subcritical periodic pattern coexisting with the sta-tionary and stable homogeneous states, and with its saddle node lying belowthe snaking region.

FIG. 10. 1D case. “Sidewinder” snaking diagram showing solid and dottedlines denote, respectively, stable and unstable multipeak configurations. Theinset is an enlargement of the diagram’s upper part. The maximum numberof CS that the “sidewinders” can accommodate is linked to the size of thespatial box chosen for the numerical simulations. Since we consider a quitenarrow box, we were limited to structures with eight or fewer peaks toobserve only 8 CS. Parameters: �=−1.0, C=5.4, and =0.25.

FIG. 11. Time plot showing the sequential switching of localized structuresby increasing the input field �parametrized by I�. In particular, for t�700 wehave I=1.578, for 700� t�1400 we have I=1.648, and finally for 1400� t�2100 the value of I is 1.713. We start from an LS1 initial conditioncorresponding to I=1.547. The color scale used goes from black to white forincreasing intensity values. Parameters: �=−1.0, C=5.4, and =0.25.

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inducing a lateral spatial shift of the centroid of the structure.This can occur on positive-slope sections of the sidewinder,as is evident in Fig. 10 and in the simulation in Fig. 11.

V. DISCUSSION

In this paper, we have identified a qualitative discrep-ancy between the theory of homoclinic snaking and relatedexperimental observations, in that localized states commonlyappear spontaneously in experiment whereas in theory theyshould only be accessible through hard excitation. We havesuggested that a competing quasiglobal nonlinearity could beresponsible for this discrepancy, and have given preliminaryevidence in favor of our conjecture. Our initial investigationshave been limited to the global coupling case, leaving forfuture work extension to the more general case of a finite-range nonlocal kernel.

Global coupling cannot produce a distance-dependentforce between LS, and so the clustering dynamics respon-sible for snaking is preserved, i.e., the snakes are tilted, notdestroyed. Nonlocal coupling, with a finite range kernel in�5�, can lead to forces that break up clusters.43

Truly global coupling is unrealistic, of course, but maywell be a good guide to the effects of finite-range nonlocal-ity. Nonlocal nonlinearity is of great current interest in itsown right, and experimental observations of nonlocal re-sponse have been reported in various systems, such as pho-torefractive crystals,44 nematic liquid crystals,45 and Bose-Einstein condensates.46 From a theoretical point of view, theeffect of a nonlocal nonlinear term, based on phenomeno-logical expressions for the integral kernel, has been an objectof study, e.g., in systems described by a NLS;47 in this case,its significant effects on the existence and stability of 2Dlocalized solutions have been demonstrated. There has beenvery little work, however, on the combined effects of localand nonlocal nonlinearity.

There are a number of physical mechanisms that couldbe responsible for nonlocality in relevant optical experi-

ments. These include transverse carrier diffusion,48 thermaleffects, and energy balances.

In the bulk semiconductor model discussed earlier, theentire nonlinearity arises from the photocarrier-field cou-pling, and so it is intrinsically nonlocal. We found a typicalquasilocal behavior, with no tilt of the homoclinic snakes,though the diffusion length was small compared to the dif-fraction length: it is not clear whether stronger diffusionalone would induce tilting. We do have preliminary evidencethat adding global coupling causes tilting just as in thesimple saturable absorber model.

Thermal inhomogeneities arising from the presence oflocalized states can be expected to induce long-range effects,and are perhaps the strongest candidate to induce sidewindersnaking. It is not yet clear whether the size, speed, or sign ofthe effects would be such as to induce tilting. Other candi-dates include boundary-induced constraints and conserva-tions laws, since they introduce a quasiglobal coupling be-tween different points of spatially extended physical systems.We note that pattern-forming systems typically involve a bal-ance between supply and dissipation of energy, and that en-ergy usually derives from a single, i.e., global, source. Thesupply of energy across the system, uniform in theory, couldbecome nonuniform in practice through intrasystem compe-tition in presence of localized states. This could lead, perhapsindirectly, to mutual inhibition of LS.

In summary, subcritical homoclinic snaking is predictedby a powerful and attractive theory, but experimental evi-dence is limited and some observations are in contradictionwith the theory. We believe that an additional global, orquasiglobal, coupling could resolve such contradictions.

ACKNOWLEDGMENTS

This work was supported in part by the EU STREP Fun-FACS, and we are grateful to all partners for discussions andaccess to results. We thank H.-G. Purwins for the provisionof material, and we acknowledge helpful interactions withG.-L. Oppo and A. J. Scroggie.

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