Tropical geometric interpretation of ultradiscrete singularity confinement

0.15 MB·English

by  Christopher M. Ormerod

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 regardingsingularitiessuchasthePainlev´epropertyandsingularityconfinement[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 Painlev´e
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,Painlev´eproperty,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 ra-
tional 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 Painlev´e
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 Painlev´e 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 equiva-
lent 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 equa-
tions as it is presented originally by Ramani et al.[1]. The singularity confinement prop-
erty 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 singular-
ity 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 singu-
larity 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 al-
though this correspondence works for this example, and indeed for all the ultradiscrete
Painlev´e 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 isomor-
phism, 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
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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
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Department of Mathematics and Statistics, The University of Melbourne, Parkville, Vic
3010,Australia.
E-mail address: [email protected]