Table Of Content

TROPICAL GEOMETRIC INTERPRETATION OF ULTRADISCRETE SINGULARITY CONFINEMENT 8 CHRISTOPHERM.ORMEROD 0 0 2 Abstract. We present a new interpretation of the ultradiscretization procedure in termsofanon-archimedeanvaluationonthefieldofPuiseuxseries. Weshalldemon- b strate how the new interpretation of the ultradiscretization gives a correspondence e F between singularity confinement for discrete mappings and ultradiscrete singularity confinement forultradiscretemappingsusingresultsfromtropicalgeometry. 4 1 Integrablecellularautomataaredynamicalsystemsthatarediscreteintime,spaceand ] h state that possess properties typically associated with integrability[17]. These properties p includetheexistenceofLax-pairs,theexistenceofconservedquantitiesandkeyproperties - h regardingsingularitiessuchasthePainlevépropertyandsingularityconfinement[17]. This t study considers ultradiscrete systems defined by a tropically rational map a m (0.1) X =Φ(X ). n+1 n [ By tropicallyrationalwe meanthe functions are expressiblesolely in terms ofthe binary 1 operations ⊕(max), ⊗(+) and ⊘(−). We consider X to be an element of Sm, where v n S=R∪{−∞} is the max-plus semiring[9]. We consider systems of the form (0.1) to be 9 5 extended cellular automata because the operations of the max-plus semiring allow the of 9 thepossiblerestrictionofX toZm ⊂Sm[15]. Thispossiblerestrictionoftheseintegrable n 1 systemstotheintegersmakesthemparticularlyattractiveinthesensethat,despitetheir . 2 simplicity, represent the very core characteristics of integrable systems[11]. 0 The ultradiscretization procedure is a routine procedure that allows one to take a 8 discrete dynamical system defined in terms of a single rational mapping, 0 : (0.2) x =φ(x ), v n+1 n i X which is describable completely in terms of +, × and ÷ (subtraction-free), to one of the form (0.1) in a way that is said to preserve integrability[11]. Ultradiscretization was r a first introduced to relate the Box and Ball system[14] to integrable lattice equations[16]. Various notions of integrability for discrete equations are naturally mapped to analogous notions in the semiring setting, such as conservedquantities[10] and the existence of Lax pairs[5]. Singularity confinement is widely regarded as a discrete analogue of the Painlevé property[1]. Recently, an analogous notion of singularity confinement has appeared for systems of the form (0.1)[3, 4]. However, singularity confinement does not naturally map via the currentform of ultradiscretizationto ananalogousnotionfor systems of the form(0.1). 2000 Mathematics Subject Classification. PACSnumbers: 05.45.-a,02.90.+p. Keywords and phrases. Ultradiscrete,Painlevéproperty,singularityconfinement . 1 2 CHRISTOPHERM.ORMEROD Wepresentanalternateinterpretationofultradiscretizationintermsofanon-archimedean valuation field. Recent advances in tropical geometry will allow us to give evidence that this new interpretation of ultradiscretizationmaps singularity confinement from systems of the form (0.2) to the recently proposed notion of singularity confinement for systems of the form (0.1). In §1 we shall present the alternate form of ultradiscretization. In §2 we shall describe the correspondence between the two forms of singularity confinement. 1. Extended interpretation of ultradiscretization We describe the ultradiscretization procedure as follows: given a subtraction-free rational function, f(x), where x = (x ,...x ) and the x s are all positive, we introduce 1 m i corresponding ultradiscrete variables, X = (X ,...,X ), via the relation x = eXi/ǫ. 1 m i The ultradiscretization of f is the tropically rational function, F, defined by the limit (1.1) F(X)= lim ǫlogf(x). ǫ→0+ To see exactly how this procedure works, let us consider how ultradiscretization applies to binary operators: (1.2a) x +x → max(X ,X ) 1 2 1 2 (1.2b) x x → X +X 1 2 1 2 (1.2c) x /x → X −X , 1 2 1 2 where 0 → −∞ and 1 → 0. From the definition given in (1.1), it is clear that if one considersthe ultradiscretizationofx −x , if X >X , then by the definition above,the 1 2 2 1 ultradiscretizationrequiresthelogarithmofanegativenumber. TheworkofKasmanand Lafortune provides methods for overcoming the difficulty presented by the minus sign in some cases[6]. This correspondence given in (1.2) motivates the following lemma. Lemma 1.1. The ultradiscretization defined by (1.1) of any (subtraction-free) rational function is equivalent to the replacement of the x with X and the replacement of binary i i operations defined by (1.2). Proof. The replacement of the x with X can be seen from the following calculation i i x → lim ǫlogeXi/ǫ = lim X =X . i i i ǫ→0+ ǫ→0+ To see that the ultradiscretization procedure is equivalent to the replacement of binary operations described by (1.2), we shall assume the ultradiscretizationof f(x) and g(x) is F(X) and G(X) respectively. By definition f(x)g(x)→ lim ǫlogf(x)g(x)= lim ǫ(logf(x)+logg(x))=F(X)+G(X) x→0+ x→0+ and f(x)/g(x)→ lim ǫlogf(x)/g(x)= lim ǫ(logf(x)−logg(x))=F(X)−G(X) x→0+ x→0+ by the properties of the logarithm. This gives correspondences (1.2b) and (1.2c). To see (1.2a), we note that by definition of the limit, for δ >0 there exists an ǫ such that 0 |ǫlogf(x)−F(X)|<δ |ǫlogg(x)−G(X)|<δ TROPICAL GEOMETRIC INTERPRETATION OF ULTRADISCRETE SINGULARITY CONFINEMENT3 for 0<ǫ<ǫ , therefor, under the same assumptions 0 e(F(X)−δ)/ǫ <f(x)<e(F(X)+δ)/ǫ e(G(X)−δ)/ǫ <g(x)<e(G(X)+δ)/ǫ. By adding these inequalities we have e−δ/ǫ(eF(X)/ǫ+eG(X)/ǫ)<f(x)+g(x)<eδ/ǫ(eF(X)/ǫ+eG(X)/ǫ) hence |ǫlog(f(x)+g(x))−ǫlog(eF(X)/ǫ+eG(X)/ǫ)|<δ for0<ǫ<ǫ . Bytakinglimitsasǫ→0givestheultradiscrtizationoff(x)+g(x)forthe 0 first term inside the absolute value signs. For the second term inside the absolute value signs, since F(X) and G(X) do not depend on ǫ, if F(X) is greaterthan G(X), then the term inside the log is asymptotic to eF(X)/ǫ as ǫ→0+, hence ǫlog(eF(X)/ǫ+eG(X)/ǫ)→ F(X), and vise versa for G(X). Since δ > 0 was arbitrary, the ultradiscretization of f(x)+g(x) is max(F(X),G(X)). (cid:3) From this, it is a routine procedure to obtain a mapping of the form (0.1) from a mapping of the form (0.2). Since our interest lies in integrable cellular automata, we consider a simple difference equation, (1.3) wn−1wnσwn+1 =atnwn+1 where |q| =6 1, t = t qn and a is constant in n. This equation is considered integrable n 0 for σ ∈ {0,1,2}. Under these conditions, this is a discrete version of the first Painlevé transcendant[11]. By letting xn = wn−1 and yn by wn, we obtain a system of the form (0.2) given by xn+1 xn yn yn+1=φyn=atxnnyynnσ+1 tn+1 tn  qtn  Since the power of x is an integer,both sides expressible solely in terms of +, × and ÷, n hence is subtraction-free. By replacing a with A, q with Q, t with T ,x by X and n n n n y by Y , then replacing binary operators on the left and right hand side in accordance n n with (1.2), we obtain the ultradiscrete mapping of the form (0.1) X X Y n+1 n n Yn+1=ΦYn=max(A+Tn+Yn,0)−Xn−σYn T T T +Q n+1 n n       or as a second order equation (1.4) Wn+1+σWn+Wn−1 =max(A+Tn+Wn,0) where which we now call the ultradiscretized version of the first Painlevé transcendant. We now presentthe alternateultradiscretizationas follows. Givena rationalfunction, f(x),wherex=(x ,...,x ),weletthex bealgebraicfunctionsinz givenbyx =z−Xi 1 n i i where the X are rational numbers. Since f(x) is rational in the x , f(x) is algebraic in i i z, hence possesses a Puiseux series representation, given by f(x)=a zq0 +a zq1 +..., 0 1 4 CHRISTOPHERM.ORMEROD where q <q for all i≥0 and a 6=0. Given this information, f(x) possesses the new i i+1 0 ultradiscretization, F(X), given by (1.5) f(x)→−q 0 Lemma1.2. The ultradiscretization of any(subtraction-free)rational function is equivalent to the replacement of the x with X and the replacement of binary operations defined i i by (1.2). Proof. Since X is rational, x is it’s own Puiseux series representation, hence, by (1.5), i i x →X . We also note that if f(x)→ and g(x) are Puiseux series given by i i f(x)=a z−F(X)+... 0 g(x)=b z−G(X)+... 0 then f(x)g(x) = z−F(X)−G(X) + ..., hence f(x)g(x) → F(X) + G(X) and similarly f(x)/g(x) → F(X)−G(X), giving (1.2b) and (1.2c). To see (1.2a), the addition of the two functions is f(x)+g(x)=a z−F(X)+b z−G(X)+.... 0 0 The ultradiscretization of this is −min(−F(X),−G(X))=max(F(X),G(X)). (cid:3) Thisallowsustodeducethatthetwoformsofultradiscretizationagreeforsubtraction freefunctions. However,inthegeneralcase,oneisnotpresentedwiththesamedifficulties in considering the definition of the ultradiscretization of general (not subtraction-free) rational functions. We denote the new ultradiscretization mapping ν :C[[t]]→S, where by C[[t]], we mean set of algebraic functions in t. In general, ν satisfies the properties 1. ν(f)=−∞ if and only if f =0. 2. ν(fg)=ν(f)+ν(g). 3. ν(f +g)≤ν(f)+ν(g). which implies ν is a valuation. However, ν also satisfies the following 3a. ν(f +g)≤max(ν(f),ν(g)). This property implies that ν is a non-archimedeanvaluation1. Hence, we have expressed theultradiscretizationprocedureintermsofanon-archimedeanvaluation. Itisimportant to note that given the two leading terms in the Puissuex series of f and g in f +g do not cancel, we have that equality holds in property (3a) of valuations. This allows us to map solutions of discrete systems that are not sub-traction free to solutions to their corresponding ultradiscrete systems[8]. Given a system such as (1.3), one may easily lift the equation to the level of algebraic functionsinz. Wesimplymakethe substitutionofw withW andconstants,{a },with n n i {zAi}. For example, a lifted version of (1.3) is simply Wn+1WnσWn−1 =zATnWn+1 where T = zQT . Given W is algebraic in z, by assigning a specific value of z, we n+1 n n recover a sequence that satisfies (1.3). Furthermore, given there is no cancellation of the dominant powers of z in the calculation of W so that equality in property (3a) of n valuations hold for all n, we recover a sequence that satisfies (1.4). 1This definition differs slightly from algebraic texts such as Lang[7]. To obtain a non-archimedean valuationinthesenseofLang,onesimplyisrequiredtoconsidereν ratherthanν itself. TROPICAL GEOMETRIC INTERPRETATION OF ULTRADISCRETE SINGULARITY CONFINEMENT5 2. Correspondence between the singularity confinements Letus first describe the singularityconfinementpropertyfor ordinarydifference equations as it is presented originally by Ramani et al.[1]. The singularity confinement property requires that given all singularities, either roots or poles, of a system, that in some limiting sense, those singularities do not propagate for all time. We present a simple ex- ampletoillustratethis definition;wepresentsingularityconfinementfortheautonomous versionof equation(1.3), where a=1, t =1, q =1 and σ =0 for simplicity. That is, we 0 consider the discrete dynamical system defined by the equation (2.1) wn−1wn+1 =wn+1. It is clear that we may write this system in the form xn+1 =φ xn = yn (cid:18)yn+1(cid:19) (cid:18)yn(cid:19) (cid:18)1+xnyn(cid:19) as one can easily see that y satisfies the same difference equation as w . Notice that in n n this special case, if w is free and w = ǫ, then we encounter a problem in defining w 0 1 3 in the limit as ǫ → 0. To see whether this system possesses the singularity confinement property, we consider the behavior of this system with the above initial conditions. The iteration of the initial conditions where w is free and w is ǫ, along with the limit as 0 1 ǫ→0 is given by w ǫ 1+ǫ 1+ǫ+w0 1+w0 w (cid:18) ǫ0(cid:19)→(cid:18)1w+0ǫ(cid:19)→(cid:18)1+ǫwǫw+00w0(cid:19)→(cid:18) 1ǫ+wǫw00 (cid:19)→(cid:18) wǫ0 (cid:19)→(cid:18) ǫ0(cid:19) limǫ→0 w 0 1 ∞ ∞ w (cid:18) 00(cid:19)→(cid:18)w10(cid:19)→(cid:18)∞w0(cid:19)→(cid:18)∞(cid:19)→(cid:18)w0(cid:19)→(cid:18) 00(cid:19) Theimportantfeatureisthatdespiteiteratingthroughasingularity,weareabletoretain information about the initial condition in the limit as ǫ → 0. We can sensibly define φ5 at (w ,0)in a limiting sense since the point (w ,0) is a removable singularity. Hence, we 0 0 say that this singularity is confined. Notice that we obtain similar problems in the case where w is free and w =ǫ−1 as 0 1 we let ǫ→0. We examine the iteration of these initial conditions: w0 → ǫ−1 → wǫ0 → wǫ0+(ǫw−01) → 1ǫ+−w10 → w0 (cid:18)ǫ−1(cid:19) (cid:18) wǫ0 (cid:19) (cid:18)wǫ0+(ǫw−01)(cid:19) (cid:18) 1ǫ+−w10 (cid:19) (cid:18) w0 (cid:19) (cid:18)ǫ−1(cid:19) limǫ→0 w −1 0 −1 −1−w w 0 → → → → 0 → 0 (cid:18)−1(cid:19) (cid:18) 0 (cid:19) (cid:18)−1(cid:19) (cid:18)−1−w0(cid:19) (cid:18) w0 (cid:19) (cid:18)−1(cid:19) The importantfeature isthat (w ,−1)is a removablesingularityofφ3. Another pointof 0 interestwhencalculatingthismapprojectivelyareistheinitialconditions(w ,∞)which 0 can be seen by letting ǫ → ∞ in the first example. In all these examples, it should be clear that the singularities are confined, since φ5 is equivalent to the identity map. Definition 2.1. A mapping, φ : S → S, is said to have the singularity confinement property if for every singularity, p ∈ S, there exists an n such that p is a removable singularity of φn. 6 CHRISTOPHERM.ORMEROD Onekeypointintheaboveexampleisthatφ5 isequivalenttoanisomorphism,despite thefactthatφ3 isnot. Thisisseenbythewaythatφ3 sends(w ,0)to(∞,∞)regardless 0 of the value of w . We may say that a whole copy of P has been sent to (∞,∞). 0 1 In this case, we may consider the blow-up of P2 at points P = (−1,0),P = (0,−1) 1 1 2 and P =(∞,∞) resulting in the surface S ⊂P5, given by 3 1 x+1 x y S = (x,y,p = ,p = ,p = . 1 2 3 (cid:26) y y+1 x(cid:27) ′ ′ If we lift φ to the mapping φ : S → S, φ is an isomorphism. The resulting surface is a highly degenerate (and simplified) form of the much celebrated surface of initial conditions[13]. We describe the ultradiscrete analogof singularityconfinement in a similar manner to discrete mappings. The ultradiscrete singularity confinement requires that any points of non-differentiability do not propagate for all time. We present the ultradiscrete of the same simple example given above. We consider the mapping defined by the iterative scheme given by (2.2) Wn−1+Wn+1 =max(Wn,0). This may also be viewed as the 2 dimensional system defined by the mapping X X Y (2.3) n+1 =Φ n = n (cid:18)Yn+1(cid:19) (cid:18)Yn(cid:19) (cid:18)max(0,Yn)−Xn(cid:19) The mapping Φ possesses a point on non-differentiability at the point corresponding to initial conditions where W is free and W = 0. Notice that although W is not 0 1 n n W :δ =0 W :δ >0 δ <0 n n 0 W W W 0 0 0 1 δ δ δ 2 −W δ−W −W 0 0 0 3 max(0,−W ) max(−δ,−W ) max(0,−W )−δ 0 0 0 4 max(0,W ) max(0,W )−δ max(0,W )−δ 0 0 0 5 W W W 0 0 0 6 δ δ δ Table 1. We compute the iterates of W when W is free and W =δ n 0 n where |δ| is a assumed to be small. This allows us to calculate the left and right derivatives at W =0. 1 differentiable at n = 2 and 3 at W = 0, W is differentiable at W = 0. That is that 1 4 1 points of non-differentiability do not propagate for all time. Definition 2.2. A mapping, Φ : S → S, is said to have the ultradiscrete singularity confinement property if, for every point, p, such that Φ is not differentiable at p, there exists an n such that Φn is differentiable at p. Just as in the discrete case, the singularities of (2.3) are confined in the ultradiscrete sense since Φ5 is the identity map. TROPICAL GEOMETRIC INTERPRETATION OF ULTRADISCRETE SINGULARITY CONFINEMENT7 Lemma 2.3. Given a subtraction free rational function, f(x) = f0(x), with ultradis- f1(x) cretization, F(X) = F (X)−F (X), the image of the roots of f and f are points of 0 1 1 2 non-differentiability of F and F . 0 1 Proof. We simply remark that this is a reformulation of Theorem 2.7 of [12]. Given a subtraction free polynomial, f , the ultradiscretization, F , the algebraic function in t 0 0 provided in the proof of Theorem 2.7 of [12] is the algebraic function in z considered in the alternate ultradiscretization obtained by replacing the variables a and x with zAi i and zx respectively. It is clear that the points of non-differentiability of the valuation considered in [12] is in one-to-one correspondence with the points of non-differentiability of the valuation considered in the alternate ultradiscretization in light of the relation max(X ,...,X )=−min(−X ,...,−X ). Finally, by the properties of the valuation, 1 n 1 n ν(f(zx))=ν(f (zx))−ν(f (zx))=F −F , 0 1 0 1 hence the points of non-differentiability is the union of the points of non-differentiability of F and F . (cid:3) 0 1 We examine the effect of the ultradiscretization of singularity confinement of over the systemdefinedby (2.1)afterlifting the systemto thefieldofalgebraicfunctions inz and applying valuation. That is we consider the system defined by (2.4) Wn+1Wn−1 =Wn+1. where the variables W are rational functions of z. To consider singularity confinement, n we have a few points of interest: W = z−∞ = 0 and W = −z0 = −1 where in both 1 1 cases, we will consider W0 to be of the form zW0 where W0 is free. n W W Root/Poles in W ND Points in W n n 1 1 0 W W ∅ ∅ 0 0 1 W W {0} {−∞} 1 1 2 1+W1 max(0,W )−W {−1} {0} W 1 0 3 1+W00+W1 max(0,W ,W )−W −W {−1−W ,0} {max(0,W ),−∞} W W 0 1 0 1 0 0 4 1+0W01 max(0,W )−W {0} {−∞} W 0 1 5 W1 W ∅ ∅ 0 0 6 W W {0} {−∞} 1 1 Table2. TheiteratesofW andW aswellasthepolesandrootsofW n n n as a function of W and the points of non-differentiability (ND Points) 1 as a function of W . 1 By considering table 2, one is use Lemma 2.3 to determine when one would expect pointsofnon-differentiabilityfromtheRootsandPolesofW asfunctionsofW alone. For n 1 example,at W =0, we see this is ν(−1), hence one expects thatW is notdifferentiable 1 2 at 0 as a function of W . Also if W < 0, then W will also not be differentiable in W 1 0 3 1 since if W < 0 then ν(1+W )=0. In the case W =−∞, we consider this to be ν(0), 0 0 0 hence we expect this to not be integrable at n=1,3 and 4. In the first case, we consider an initial condition where W is close to 0, namely an 1 initial condition where W = z−L where we consider W → 0 as L → 0. Hence, we may 1 1 8 CHRISTOPHERM.ORMEROD consider the valuation under the assumption that L≫0. n W W =ν(W ):L≫0 n n n 0 zW0 W 0 1 z−L −L 2 z−W0(1+z−L) −W0 . 3 z−W0(1+zL+zL+W0) L−W0+max(W0,0) 4 zL(1+zW0) L+max(0,W ) 0 5 zW0 W0 6 z−L −L The first feature of this is that due to the subtraction-free nature of W as rational n functions of z with these initial conditions W = ν(W ) satisfies (0.1). The second n n feature is that, as predicted by Lemma 2.3, the mapping is not differentiable at n=1,3 and 4 as the functions diverge. Also notice that by assigning a value to z, as L → ∞, z−L →0 or ∞. In either case, what is observed is singularity confinement for (0.2). WenowturntotheinitialconditionaroundW =−z0,givenby−zδ where|δ|≪|W |. 1 0 There are two cases we are required to consider: the case in which δ < 0 and the case where δ >0. The valuation of W for each n depends on δ. n n W W =ν(W ):δ >0 W =ν(W ):δ <0 n n n n n 0 zW0 W0 W0 1 −zδ δ δ 2 z−W0(1−zδ) −W −δ−W 0 0 . 3 z−δ−W0(zδ−1−zW0) −W0+δ+max(0,W0) −W0+max(0,W0+δ) 4 −z−δ(1+zW0) δ+max(0,W0) δ+max(0,W0) 5 zW0 W0 W0 6 −zδ δ δ The firstfeature, aswas observedin the previouscase,is thatW =ν(W ) satisfies(0.1) n n for cases in which δ > 0 and δ < 0. This can be guaranteed by sufficiently generic (and generally sufficiently small) values of δ so that there is no cancellation of highest power terms of z in the calculation of W . Furthermore, as predicted by the use of Lemma 2.3, n the function is not differentiable at n=2, while for n=3, W is differentiable if W <0 n 0 and is not differentiable if W ≥0. 0 As remarked at the end of §2, the correspondence between the two types of singularity confinement can be summarized as follows. After lifting the equation to the level of algebraic functions in a variable z, by assigning a specific value of z, we recover singularity confinement as it is defined for discrete mappings. However,givenone takes initial conditionssufficiently andgenericallyclosetothe singularities,onerecoversultradiscrete singularity confinement via the application of the valuation. 3. Discussion We presented a very simple autonomous example to demonstrate the correspondence between the two types of singularity confinement. We could have chosen any one of the equations presented in [11], however, the system chosen was one of the simplest systems we could find that displayed all the desired characteristics. It is worth noting that although this correspondence works for this example, and indeed for all the ultradiscrete Painlevé equations that featured in [11], what we have demonstrated is not a one-to-one TROPICAL GEOMETRIC INTERPRETATION OF ULTRADISCRETE SINGULARITY CONFINEMENT9 correspondence between the two singularities. In fact the inverse of the valuations of hopelessly multi-valued. A case in which we have not ruled out is one in which by iter- ating out of one singularity, you are in the image of another pole or root. Furthermore, we have provided a surface of initial conditions in which the mapping φ was an isomorphism, however, further study is required to determine what the analogous notion is for ultradiscrete systems such as Φ. 4. Acknowledgements IwouldliketoacknowledgeProfessorKenjiKajiwaraforhisencouragementandhelpful discussions. As well as Professor Reinout Quispel for his encouragement. The research of C. M. Ormerod was supported in part by the Australian Research Council Grant #DP0559019. References [1] Grammaticos B, Ramani A and Papageorgiou V 1991 Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67(14) 1825–1828 [2] Grammaticos B, Ramani A, Tamizhmani K M,Tamizhmani T and Carstea A S2007 Dointegrable cellularautomata havetheconfinement property? J. Phys. A: Math. Theor. 40 (30)F725F735 [3] Joshi N and Lafortune S 2005 How to detect integrability in cellular automata J. Phys. A 38 (28) L499–L504 [4] Joshi N Lafortune S 2006 Integrable ultra-discrete equations and singularity analysis. Nonlinearity 19(6) 1295–1312 [5] JoshiN,NijhoffF,OrmerodCM2004Laxpairsforultra-discretePainlevécellularautomata J.Physc A: Math.Gen. 37(44) L559–L565 [6] Kasman A and Lafortune S 2006 When is negativity not a problem for the ultradiscrete limit? J. Math. Phys. 47(10) 103510 [7] LangS1965Algebra Addison-Wesley Publishing Co., Inc., Reading, Mass. [8] OrmerodCHypergeometricsolutionstotheultradiscretePainlevéequations In preparation [9] PinJ1998Tropicalsemi-rings,Idempotency Publ. Newton Inst. 11 50–69 [10] Quispel G R W, Capel H W, Scully J 2001 Piecewise-linear soliton equations and piecewise-linear integrablemaps J. Phys. A 34(11) 2491–2503 [11] RamaniA,TakahashiD,GrammaticosBandOhtaY1998TheultimatediscretisationofthePainlevé equations Phys. D 114(3-4) 185–196 [12] Richter-Gebert J, Sturmfels B and Theobald T 2005 First steps in tropical geometry Idempotent mathematics and mathematical physics Contemp. Math. 377 289–317 [13] Sakai H 2001 Rational surfaces associated with affine root systems and geometry of the Painlevé equations Comm. Math. Phys. 220 165–229 [14] TakahashiD,SatsumaJ1990Asolitoncellularautomaton J.Phys.Soc.Japan 59(10) 3514–3519 [15] TokihiroT2004Ultradiscretesystems(cellularautomata).Discreteintegrablesystems LectureNotes in Phys. SpringerBerlin 644 383–424 [16] TokihiroT,TakahashiD,MatsukidairaJandSatsumaJ1996FromSolitonEquationstoIntegrable CellularAutomatathroughaLimitingProcedure Phys. Rev. Lett. 76(18) 3247–3250 [17] ZakharovVE(editor)1991Whatisintegrability? Springer-Verlag, Berlin Department of Mathematics and Statistics, The University of Melbourne, Parkville, Vic 3010,Australia. E-mail address: [email protected]