Physics LettersA 169 (1992) 151—160 PHYSICS LETTERS A North-Holland Integrable cellular automata M. Brusehi, P.M. Santini and 0. Ragnisco Dipartimento di Fisica, Università “La Sapienza”~ 00185 Rome, Italy and INFN, Sezione di Roma, Rome, Italy Received 27 February 1992; revised manuscript received 21 July 1992; accepted for publication 22 July 1992 Communicated by A.P. Fordy We present a general procedure to associate hierarchies of cellular automata with given spectral problems choosing, asillustra- tive example, the discrete Schrodinger problem. For these cellular automata we construct a countable number of constants of motion using standard spectral techniques and we perform numerical experiments showing interesting dynamical features and particle content. In the second part ofthe paper we introduce other cellular automata characterized by a very rich particle content and by the existence of constants of motion. 1. Introduction 1 + 1 dimensional cellular automata (CA) are dynamical systems in the discrete space variable neZ and time variable tel, in which the field variable q=q(n, 1) takes values on a finite field, say {O, 1, ..., m— l}, mEl’J. CA exhibit a very large variety of dynamics and empirical attempts of classification have been made [1]. A new class of 1 + 1 dimensional CA, called parity-rule filter automata (PRFA) (see section 2 for the def- inition), has been introduced in ref. [2] several years ago. These PRFA exhibit a vast array of coherent struc- tures, called particles, which in many cases do interact solitonically. The discovery of a “fast rule” [3], an effective method of solving the PRFA similar, in spirit, to the hodograph transformation, has allowed one to investigate analytically the rich properties of the solitonic structures [4]; see also ref. [5]. Generalizations to states taking values in finite groups were obtained in ref. [6]; variants of the above PRFA were introduced in ref. [7]. In a recent paper [8] a class of time reversible CA, which incorporate the most interesting features of the above PRFA, was proposed. In this work the issues of time reversibility and stability were fully addressed. These CA possess also conservation laws [9], therefore they share with soliton equations many of the relevant properties of integrabiity. We do not know any other results in the direction of a mathematical foundation of the notion of integrability for CA; in particular the problem of constructing CA as integrability conditions for linear operators (the well- known Lax pair [10]) is, to the best of our knowledge, open. To the latter problem we devote the first and main part of this work (see section 2), in which: (i) we introduce a general procedure to generate hierarchies of integrable CA starting from given spectral problems. For example we show that the following filter, reversible, 1+ 1 dimensional CA, q(n, t+l)=q(n, t)+q(n—2, t+l)q(n+ 1, t)+q(n—l, t+1)q(n+2, I) , mod2, (1) q(n,t+1)=q(n,t)+q(n—2,t+1)q(n+3,t)[l+q(n—l,t+l)}[l+q(n+2,t)] +q(n—3,t+1)q(n+2,t)[1+q(n—2,t+l)][1+q(n+l,t)], mod2, (2) are the first two members of a hierarchy of CA associated with the discrete Schrodinger spectral problem [11] Elsevier Science Publishers B.V. 151
1 + 1 dimensionalcellularautomata(CA) aredynamicalsystemsin the discretespacevariableneZandtimevariabletel, in which the field variableq=q(n, 1) takesvalueson a finite field, say {O, 1, ..., m— l}, mEl’J.
CA exhibit a verylargevarietyof dynamicsandempiricalattemptsof classificationhavebeenmade[1].A newclassof 1 + 1 dimensionalCA, calledparity-rule filter automata(PRFA) (seesection2 for the def-
inition), hasbeenintroducedin ref. [2] severalyearsago.ThesePRFAexhibit a vastarrayofcoherentstruc-tures,calledparticles,which in many casesdo interactsolitonically. The discoveryof a “fast rule” [3], aneffectivemethodof solvingthe PRFA similar, in spirit, to the hodographtransformation,hasallowedonetoinvestigateanalyticallythe rich propertiesof the solitonicstructures[4]; seealso ref. [5]. Generalizationstostatestaking valuesin finite groupswereobtainedin ref. [6]; variantsof the abovePRFA were introducedin ref. [7].
In a recentpaper [8] a classof time reversibleCA, which incorporatethe mostinterestingfeaturesof theabovePRFA, wasproposed.In this work the issuesof time reversibility and stability werefully addressed.TheseCA possessalsoconservationlaws [9], thereforetheysharewith solitonequationsmanyof the relevantpropertiesof integrabiity.
Wedo notknow anyotherresultsin thedirectionofa mathematicalfoundationofthenotionof integrabilityfor CA; in particulartheproblemof constructingCA asintegrabilityconditionsfor linearoperators(the well-known Lax pair [10]) is, to the bestof our knowledge,open.To the latterproblemwe devotethe first andmain part of this work (seesection2), in which:
(i) we introducea generalprocedureto generatehierarchiesof integrableCA startingfrom givenspectralproblems.For examplewe show that the following filter, reversible,1+ 1 dimensionalCA,
(ii) we constructa countablenumberof constantsof motion throughelementaryspectralconsiderationsassociatedwith (3); in particularwe show that two suchconstantsof motion arejust the boundariesof thecompactsupportinitial configuration.Thereforethedynamicsof our CA takesplacestrictly insidethesecom-pactregionsandthe evolutionis stableandperiodic; indeedcompletelyperiodicfor the subclassof time re-versibleCA (seefigs. la—id);
(iii) weperformnumericalexperimentsshowingthatgenericlocalizedperturbationsoforderedbackgroundsinside (arbitrary large)compactregionstravel with velocities0, ±1, ~, interactsolitonically (possiblywithcreationandannihilationphenomena)andare reflectedby thefixed boundariesof the support(seefigs. 2a—2c). Theanalogieswith the motion of “dislocations”in solid physicsandwith the “quark confinement”inparticlephysicsare suggestiveandgive to the abovelocalizedperturbationsa particle-like content;thereforewe will often use the word particlesto denotethem (seesection2.2).
The secondpart (section3) of thisLetter is different, in spirit, from the first oneandis closerto ref. [8].In this partwe introducea classof filter, quadratic,reversibleandstableCA, characterizedby a veryrich par-ticle dynamicsandby the existenceof constantsof motion.Althoughfor theseCA we haveup to now neitheran integrability scheme(like the one introducedin section2) nora fast rule [31, neverthelesswe think it isworthwhile to presentthem,giventhe greatrichnessof their dynamics(seefigs. 3a—3d).
2. Integrablecellular automata
2.1. Integrability schemeandconstantsof motion
We first show how to generateCA from lineardifferencespectralproblemsandwe consider,as prototypeexample,the discreteSchrbdingerproblem (3). Assumingthat w andq dependon the discretetimevariabletel, we seeka compatibletimeevolutionin the form
yJ(n,t+l)=~Vw(n,t)w(n+2j,t;z), MeN. (4)
If M= 1, compatibility between(3) and (4) yields
V~°~(n—1, t) = V~°~(n,1) =const=1, without lossofgenerality, (5a)
In orderto view eqs.(5) as definingaCA, wehaveto interpretequalitiesas“modulo congruences”,indicatedhereafterwith thesymbol“mod m”, wherethemodulusm is a primenumber.In otherwordsweseeksolutionsof eqs.(5) in the finite field Zm:= Z/mZ = {0, 1, ..., m — 1 }. Equation (Sc) thenadmitsthe basicsolution
V”~(n,t)=bô(l+q(n—l,t+l))5(l+q(n+2,t)), modm, (6)
wheretheconstantbeZm,ô(x) = 0, modm, iffx~0, modm,and,in thefollowing, we will usetherepresentation
rn-I
ô(x)= fl (k+x), modm, XE7Lm. (7)k= I
Substitutingeq. (6) into eq. (Sb) we obtain the following explicit law of the CA,
whereb, k~Zmandfl(x)=>~7!~x~is the solutionof theequationxfl(x)=fl(x), modm, form prime.Sub-stituting(1 Oa) into (9b) weobtainthe secondmemberof thehierarchyof CA associatedwith (3). Form= 2,k= b= 1, the explicit form of it is given in eq. (2). It is clearhow to proceedfor anyMeN.
It is easyto showthat the aboveCA possessthe following properties:Property1. Thegeneralsolutionof (Sc) is obtainedmultiplying thebasicsolution(6) by anarbitraryfunc-
tion of q. Analogously,the generalsolution V“~of (9c) is obtainedmultiplying the basic solution (1Oa) byanarbitraryfunction of q andthenaddingto it the generalsolution of (5c) (this is the linear superpositionprinciplefor ourCA). Althoughin thisLetterwewill focusessentiallyontheCA generatedby thebasicsolution(the“basicCA”), someofthe resultsarealsovalid for theCA generatedby the generalsolution (the“generalCA”).
Property2.All thebasicCA areexplicitlycomputableforanyinitial conditionq(n, 0) on “compactsupport”(i.e. suchthat q(n,0)=0 for InI>N, for someNeIN).
Property3. ThebasicCA arefilter CA [2], sincethevalueofq(n0, t+ 1) dependsnotonly on q(n,1), ne{no,
n0+l, ..., no+r} but also on q(n, t+l), ne{no—r,n0—r+l, ..., n0—l}, wherer is the so-called“interactionradius”.
Property4. Eachmemberof the abovehierarchyof basicCA hasa differentradiusof interaction;preciselyr=M+l.
Property5. Decomposinganarbitraryinitial configurationon compactsupportin a finite numberof regionsseparatedbyatleast2r— 1 consecutivezeroes,it iseasyto showthatageneralCA inducesevolutiononlyinsidesuchregions(indeedstrictly inside, i.e. the boundaryvaluesareconstantunderthe evolution(seealso (1 6e)and(1 6h))). Thereforewecall suchregions“islands” (seefig. 1 a— 1d); notethatthispropertyimpliestriviallythe stability of theseCA (the numberof nonzerositesstaysfinite for anyfinite time).
Property6.ThebasicCA are“time-reversible”,namelytheypossessthesymmetryq(n+j, t)~—’q(n—j,t+ I).Property7. Promproperty5 it follows that generalCA are periodic (i.e. the systemattainsperiodiccon-
figurations,possiblyaftera transient).Properties5 and6 imply that the basicCA are “completelyperiodic”(i.e.any initial conditionis alreadypart of a periodicconfiguration) (seefigs. la—ld).
We stressthenovelty of our solutionof eqs. (5) and(9) viewedasmodulocongruenceswith respectto theordinarysolutionsof the sameequations.For instancethe ordinarysolution of eq. (Sc) reads
Fig. 1. In figs. 1—3 thehorizontalaxisis the n (space)axisandthedownwardverticalaxisis thet (time)axis. Differentcolors(whiteblack andgrey) stayfor differentoccupationnumbers.(a)—(d) Islandconfigurationsfor CA (8) and (9), (10). (a) CA (8) mod 2;(b) CA (8) mod 3; (c) CA (9), (10) mod 2; (d) CA (9), (10) mod 3. In someof theislandsthecompleteperiodicityis exhibited;inotherstheinitial configurationis notrestoredyet.
timeevolutiondescribedby suchBTs is differentfrom thatof thecorrespondingCA. Moreprecisely,if we letthe stateq(n, I) evolveaccordingto bothequations,wediscoverthat the stateq ( n, t + 1), evolvedaccordingto the BT, is notmodulo congruentto the stateevolvedaccordingto theCA.
The Lax equations(3), (4), viewedasmodulocongruences,shouldprovide, in principle, thelinearizationschemefor the abovehierarchyof CA. Postponingto a subsequentpapera completestudyof sucha scheme(whichpresentsseveralnovel features)hereweconcentrateon thoseaspectsof thetheorywhich aresufficientto provethat theaboveCA possessa countablenumberof constantsof motionandto constructthemexplicitly.This constructionis hereachievedfollowing two different approaches:
First approach.The aim of this approachis to show that the time evolutionsof the spectraldataassociatedwith (3) arelinearmodulocongruences.Since the spectraldataare functionsof the spectralparameterz, wefirst introducethe following definitions:
are modulocongruent:h1(z)=h2(z), modm, iffck1~=cJ~2~,modm, Vk.
(ii) Two meromorphicfunctionsg1 (z), g2(z) of z,
i=l,2, (13)
are modulo congruent:g1(z)=g2(z), mod m, iffh1(z)=h2(z), mod m, and ~c’1(z)=E2(z),mod m with~(z)~O, mod m.
Now we consideran initial configurationon the compactsupport{no, ..., n0+N— 1 } (i.e. q(n) = 0 if n <n0or n> n0+N— 1) andwe look for the solution w(n; z) of (3) behavinglike z” at n—p+co; then the scatteringdataa(z) and b(z), defined by
~+a(z)z°+b(z)z”, n—’—oc, (14)
havethe following form,
1 N-la(z)=—T———~ a~z
2~, a0=—l, (l5a)
z — 1 ~
N—I2 ~ b~z
2~, (15b)z —l ~
wherethe coefficientsa3 and b~canbe expressedin termsof q. For instance
Noting that V°~—~Oas nl-+co for j~O, eqs. (4), (5a) and (9a) show that ~v(n, t+l; z) and ~(n, t; z) areasymptotically (in n) modulo congruent; therefore
i.e. the a1 andthe b~are constantsof motion for the aboveCA.Remarks. (i) In the usualcase,evolutionsaccordingto the integrablenonlinearequationsassociatedwith
(3) (which include the Kac—van Moerbeke lattice [11]), imply the conservation of the transmissioncoeffi-cient T(z)= 1/a(z) and then of the countable number of coefficients of any suitableasymptoticexpansionofT in z. Theseconstantsof motion,beingjust combinationsof the a~,are sharedalso by our CA which, in ad-dition,exhibitsthe constantsof motionb~.Notehoweverthatnotall the constantsof motion herederivedareindependent;for instancethe conservationof (16e) and (16h) implies the conservationof (1 6d).
(ii) From the conservationof (1 6e)and (16h) it also follows that thevaluesof q (n) at theboundariesofthe supportdo not evolve,as mentionedin property5.
(iii) ContrarytothesolitonicequationsofKdV type,thelinearproblem(3) canapparentlybesolveduniquelyfor eigenfunctionsw( n; z) normalized at + ~ (aswe did aboveto generatethe constantsof motion) but notforeigenfunctionsnormalizedat — ~ since,in thiscase,oneshoulddealwith thedenominator1 +q(n), whichcouldbe zeroin 74,. This novel featureandthe possiblelimitationsof theproposedintegrabilityscheme(3),(4) will be discussedin a forthcomingpaper.
Secondapproach.This approachis basedon propertyS. Since the evolutionof a genericinitial datumsplitsinto a finite numberof independentevolutionsfor eachislandof the support,it is naturalto reformulatetheproblemfrom the line neN into a finite collection of independentproblemson eachisland.Accordinglywewill focuson a singleisland,say I={n=l, 2,...,N—l} with q(n)=O for nd andwe introducethefollowingNXNmatrix,
LJk=öJ_I,k+öJ±I,k[l+q(j,t)], j,k=l,...,N. (19)
Then it is straightforward to show that the CA (5) and (9) are equivalent to the equation
L(t+l)=AL(t)A , modm, (20)
wherethe NxNmatrix A is given, respectively,by
AJk=öJk+~J+2kV”~(j,t) (2la)
and
AJk=öJk+öJ+2kV~
1~(j,t)+ôJ+4kV~2~(j, I) , (2lb)
where,in eq. (21a), V~1~is given by eq. (6) and,in eq. (2lb), ~ V~2~are givenby eqs.(10). Equation(20) immediately shows that the coefficients c
3 of the secularequationdet(L—)J)= ~~‘~ocs)!= 0 or, equiv-alently, the tracestr(L’), j_—1, ..., N, are constantsof motion for the CA. Due to the structureof L, it turnsout that cN_(21+1) = 0, j= 0, 1, ..., [(N— 1)/2]. Thestudyof the equivalenceof the two approachesandof therelationbetweenthedifferent setsof constantsof motion {a3}, {b3} and{c3} is postponedto a subsequent paper.
2.2. Qualitativefeatures
Althoughperiodic (or evencompletelyperiodic),the dynamicsof ourCA is far from trivial. Indeedacom-puterizedinvestigationof the dynamicsin sufficiently largeislandsshowsthe following phenomena:
(1) existenceof suitableorderedbackgroundmediain which particle-like objectscanmove (seefigs. 2a—2c);
(2) two particlescaninteractcompletelysolitonically (i.e. their interactionproducesonly aphaseshift, pre-servingnumber,shapeandvelocity of the particlesinvolved) or they canproducenew particles(seefigs. 2a,2b);
(3) the particlesare reflectedby the boundariesof the island, interactingwith them as in point (2) (seefig. 2c);
~Fig. 2. Particularevolutionsinsidea single islandfor CA (8).(a)CA (8) mod2; it showsthemotionofparticlesin an orderedbackgroundfarfrom the boundariesof the island.Theparticlemovingto theright collidessuccessivelywith threedifferentpar-ticlesmovingto theleft. The first two collisionsarecompletelysolitonic, thelastcollision producesdifferentparticles.(b) CA(8) mod 2; it showsparticlesin adifferentbackgroundmoving
~ to theleft andcolliding, solitonicallyandnot, with particleswithinfinite velocity. (c) CA (8) mod 3; it describesthemotionof
(4) the only velocities empirically observed are 0, ±1, ~ (seefigs. 2a—2c). An infinite velocity of prop-agationis possiblein filter CA.
We finally remark that the configurationswhich we call islandsare referredto in the literature [1—7]asparticleswith zero velocity andinternaldynamicalstructure.Thereforewhatweobserveinsidean islandcanbe interpreted as confined particles movinginsidea steadyparticle.
2.3. Openproblems
Associatedwith the aboveintegrableCA, the following fundamentalquestionsare presentlyunderinves-
tigation andwill betreatedin a subsequentpaper:(i) how to usethe integrabiity schemes(3), (4) and/or (19)—(2l) to linearizethe CA andto describe
analyticallythe abovephenomena;(ii) how to predictanalytically the periodfrom the initial configuration;(iii) how to view these CA as completely integrable Hamiltonian systems;(iv) how to extendthis theoryto higherdimensionsin spaceandto higherderivativesin time.
Thebasic CA introducedin section2 exhibit the following structuralproperties:Propertya. Theyarepolynomial (thedegreeofthepolynomialdependingon themodulom andontheradius
r of interaction),computable,filter, time reversibleCA.Propertyb. Eachnonlinearmonomialof the CA is filter, namelyit containsboth termsq( t) andq(1+ 1).Propertyb is responsiblefor property5 of section2, namely it implies that the evolutionis restrictedto
islands.It is therefornaturalto seekCA which enjoypropertiesa without the restrictionb, in thespirit of ref.[8]. Themain problemarising in this questis stability, sincea genericfilter, polynomial, reversibleCA canyield a “vacuum excitation”, i.e. the production of aninfinite configurationfrom a finite one in just onetimestep.
Even in the simplest possible case (modulo 2 and nonlinearity of degree 2) it is possible to find CA whichpossess property a, avoiding vacuum excitations. For instance, for r= 2 and r= 3 wefound,respectively,thefollowing novel CA,
where cx3, hikE {0, 1 }. A straightforward investigation of the evolution of all possible rightmost configurationsproves the stability of the above CA. For instance, for CA (22) we proceed as follows. If, at time I, q( n, t) = 0for n>N, then q(n, 1+1), n>N, depends only on the terms q(n—l, t+l) and q(n—2, t+l)q(n—l, t+l).Therefore,at time t+ 1, the four possible configurations00, 01, 10, 11 of the sitesn—2, n — 1 (n> N) give rise,respectively, to the configurations 000..., 100..., 000..., 000... respectively of the sites n, n + 1, n + 2, ... (n>N).
TheaboveCA exhibit a very rich phenomenology,investigatedvia computer,which we summarizein thefollowing points:
(i) thereexistanenormousarrayof coherentstructures,which we call particles,travellingwith differentvelocities in both directions (see figs. 3a—3c);
(ii) any finite initial condition splits, asymptotically in time, into a finite numberof particles(seefig. 3a);(iii) the interactionbetweentwo or more particles can be completely solitonic (i.e. their interaction pro-
ducesonly a phaseshift, preservingnumber,shapeandvelocity of eachparticle),or onecanhaveproductionor gluingof particles(seefigs. 3a—3c);
(iv) it is possibleto constructvariousmedia,possiblynonhomogeneous;in thesemediatwo or morepar-ticles interactwith eachotherlike in (iii) aswell aswith the medium,exhibitingthe refractionphenomenon(see fig. 3d).
The problem of associating with the CA (22), (23) anyintegrationschemeis presentlyunderinvestigation.Herewe only remarkthe existenceof the following constantsof motion,
Fig.3. Particularevolutionsinducedby CA (22) and(23). (a) CA (23);it showstherapiddecayofa genericinitial datuminto afinitenumberof differentparticlesmoving with differentvelocities in both directions. (b) CA (23); it showssolitonic andnonsolitoniccollisionsbetweenparticles. (c) A surprisinggluingphenomenonfor CA (22) implementedby expression(24) with a
1 1, a2=/ijk=O.(d) CA (22);it showstwo particlesenteringa moving nonhomogeneousmedium; the particles undergorefractionby the medium,collidesolitonicallyandfinally getoutof themediumwith theinitial shapeandvelocity.
and
E1=>q(j), mod2, (26)feZ
for theCA (22) and(23) respectively.Theyareobtainedmultiplyingeqs.(22) and(23)by suitablefunctionsof q andthensummingoverfeZ.
We endremarkingthat alsofor the CA of this section it would be interestingto investigateextensionsinhigherspacedimensionsandin highertime derivatives.
Weacknowledgeusefuldiscussionswith S.V. Manakov.We acknowledgeusefulcommentsof M.J. Ablowitzon the first versionof the paper.This work hasbeenpartially supportedby the Italian Ministery of the Uni-versity andScientificResearch.
References
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