Ultradiscrete Painlev\'e VI with Parity Variables

We introduce a ultradiscretization with parity variables of the $q$-difference Painlev\'e VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of $q$-Painlev\'e VI satisfies the ultradiscrete Painlev\'e VI system of equations with the parity variables, which is valid by using the parity variables. We study some solutions of the ultradiscrete Riccati-type equation and those of ultradiscrete Painlev\'e VI equation.


Introduction
The Painlevé equations appear frequently in the problem of mathematical physics, and they have extremely rich structures of mathematics [1]. The q-Painlevé equations are q-difference analogues of the Painlevé equations [8], and most of them have symmetry of affine Weyl groups, which play important roles to analyze integrable systems [6,10]. On the other hand, cellular automaton has been studied actively and has been applied to vast areas of science and technology. Although some of cellular automaton describe complexity from their simple rule of evolution, some of them have integrability [12].
It is easy to confirm that the relation xy = z (resp. x/y = z) corresponds to X + Y = Z (resp. X − Y = Z). This procedure is sometimes called ultradiscretization. The addition, the multiplication and the division correspond to taking the maximum, the addition and the subtraction by the ultradiscretization. However the subtraction x − y is not well-behaved by the ultradiscretezation. Moreover if some values of x, y, z are negative, then the procedure does not work well. To overcome these troubles, Satsuma and his collaborators [4] introduced the ultradiscretization with parity variables, and they obtained the ultradiscrete Painlevé II with parity variables [2]. Then the ultradiscrete Airy function with parity variables appears as a special solution of the ultradiscrete Painlevé II with parity variables [2].
The q-difference Painlevé VI system of equations was discovered by Jimbo and Sakai [3], and it is written as z(t)z(qt) b 3 b 4 = (y(t) − ta 1 )(y(t) − ta 2 ) (y(t) − a 3 )(y(t) − a 4 ) , y(t)y(qt) a 3 a 4 = (z(qt) − tb 1 )(z(qt) − tb 2 ) with the constraint b 1 b 2 a 3 a 4 = qa 1 a 2 b 3 b 4 . The original Painlevé VI equation is recovered by the limit q → 1 (see [3] The q-Painlevé VI system has Riccati-type solutions in the special cases [3]. Namely, if b 1 a 3 = qa 1 b 3 , b 2 a 4 = a 2 b 4 and the functions y(t) and z(t) satisfy the following Riccati-type equation: then the functions y(t) and z(t) satisfy the q-Painlevé VI system. It is also known that the Riccati-type equation has solutions expressed by q-hypergeometric functions [3,9]. In this article, we consider ultradiscretization of the q-Painlevé VI system with parity variables. For each value m of the independent variable, we associate the signs y m , z m ∈ {±1} and the amplitudes Y m , Z m ∈ R. Define a parity function S(ζ) for a sign variable ζ by S(1) = 0 and S(−1) = −∞. We now introduce the ultradiscrete Painlevé VI system of equations with the variables (y m , Y m ) and (z m , Z m ) by with the constraint On equations (3), (4) we ignore the terms containing S(ζ) = −∞ in the maximum. Note that ultradiscretization of the q-Painlevé VI equation without parity variables was already introduced by Ormerod [7], and it is recovered by fixing the parity variables by y m = z m = −1. Equations (3), (4) are obtained by ultradiscretization of the q-Painlevé VI system under the condition a i > 0 and b i > 0 (i = 1, 2, 3, 4), where the condition is used in the process of obtaining the original Painlevé VI equation in [3], and we can also obtain the ultradiscrete Painlevé VI system of equations which admits the parities of the parameters (see equations (17), (18)). On the ultradiscrete Painlevé VI system of equations with parity variables, we have existence of the solution of the initial value problem, although the uniqueness does not hold true. We also ultradiscretize the Riccati-type equation with parity variables and show that any solutions of the ultradiscrete Riccati-type equation satisfy the ultradiscrete Painlevé VI system. Here the ultradiscretization with parity variables is essential because the ultradiscrete Riccati-type equation does not have any solutions in the case y m = z m = −1.
We try to study the solutions of the ultradiscrete Riccati-type equation and those of the ultradiscrete Painlevé VI equation. We give examples of solutions which are described by piecewiselinear functions. Based on numerical calculations, we present a conjecture that the solutions are expressed as linear functions if the independent variable m is enough large (for details see Conjecture 1). This paper is organized as follows. In Section 2, we consider ultradiscretization of the Riccatitype equation and that of the q-Painlevé VI equation. We establish existence of the solution of the initial value problem for the ultradiscrete Painlevé VI system of equations. In Section 3, we show that any solutions of the ultradiscrete Riccati-type equation also satisfy the ultradiscrete Painlevé VI equation. In Section 4, we investigate solutions of the ultradiscrete Riccati-type equation and those of the ultradiscrete Painlevé VI equation.

Ultradiscretization of the Riccati-type equation
We consider ultradiscretization of the Riccati-type equation at first, because the expression is simpler than those of the q-Painlevé VI equation.
To obtain the ultradiscrete limit, we set and define the parity functions s(ζ) and S(ζ) by Then we have s(ζ) = e S(ζ)/ε for ε > 0. We substitute equation (6) into the equation which is equivalent to first equation of (2), and transpose the terms to disappear the minus signs. Then we have s(z m+1 )e Z m+1 /ε s(y m )e Ym/ε + s(−z m+1 )e Z m+1 /ε s(−y m )e Ym/ε + e A 4 /ε + e B 4 /ε s(−y m )e Ym/ε + e (mQ+A 2 )/ε = e B 4 /ε s(y m )e Ym/ε + s(z m+1 )e Z m+1 /ε s(−y m )e Ym/ε + e A 4 /ε + s(−z m+1 )e Z m+1 /ε s(y m )e Ym/ε . By using the formula s(y)s(z) + s(−y)s(−z) = s(yz) and taking the limit ε → +0, we have It follows from the second equation of (2) that We call equations (7), (8) the ultradiscrete Riccati-type equation with parity variables. By the ultradiscrete limit, the conditions b 1 a 3 = qa We write the equations for the amplitude variables by fixing the parity variables. Equation (7) for each case is written as There is no solution in the case z m+1 = y m = −1. Equation (8) for each case is written as There is no solution in the case z m+1 = y m+1 = −1.

Ultradiscretization of the q-Painlevé VI equation
We can obtain similarly the ultradiscrete limit with parity variables of the q-Painlevé VI equation. We take a limit of equation (1) as ε → +0 by setting the values as equation (6). By using the formulae we obtain the ultradiscrete Painlevé VI equation with parity variables (i.e. equations (3), (4)).
In [13], another form of equations (3), (4) is derived with the details. By the ultradiscrete limit, the constraint b 1 b 2 a 3 a 4 = qa 1 a 2 b 3 b 4 of the q-Painlevé VI system corresponds to equation (5). We write the equations for the amplitude variables by fixing the parity variables. Equation (3) for each case is written as In the case y m = −1, z m z m+1 = 1, we used the associativity of the maximum and the addition, i.e.
There is no solution in the case y m = z m z m+1 = −1. On the other hand, equation (4) for each case is written as There is no solution in the case z m+1 = y m y m+1 = −1.
We can also obtain the ultradiscrete Painlevé VI system of equations which admits the parities of parameters. Set in addition to equation (6), where a i , b i ∈ {±1} represent the signs of a i , b i (i = 1, 2, 3, 4). We take the ultradiscrete limit (ε → +0). Then we have a 1 a 2 b 3 b 4 ), a 3 a 4 ), with the constraint By setting a i = b i = +1 (i = 1, 2, 3, 4), we recover equations (3), (4). In the rest of the paper, we consider the case a i = b i = +1 (i = 1, 2, 3, 4), i.e. equations (3), (4) for simplicity. The ultradiscrete Painlevé VI equation with parity variables has solutions for any give initial values. Namely we have the following proposition: Proof . We show that, if the values (z m , Z m ) and (y m , Y m ) are fixed, then there exists (z m+1 , Z m+1 ) such that equation (3) is satisfied.

then equation (3) is satisfied by setting
We proceed it for m = n o , n o + 1, . . . , and a solution (y n , Y n ), (z n , Z n ) (n > n o ) is obtained. By applying a similar procedure to equation (4) for n = n o − 1, equation (3) for n = n o − 1 and so on, a solution (y n , Y n ), (z n , Z n ) (n < n o ) is obtained. Thus we have a solution (Y n , Z n ) (n ∈ Z) which satisfies the condition (y no , Y no ) = (ỹ o , y o ) and (z no , Z no ) = (z o , z o ).
The uniqueness of the solution is not satisfied for some cases. In the case U = U or V = V in the proof of the proposition, i.e. the case Y m = A 3 , A 4 , A 1 + mQ or A 2 + mQ, we do not have uniqueness of the solution. Similarly in the case Z m+1 = B 3 , B 4 , B 1 + mQ or B 2 + mQ, we do not have uniqueness of the solution. On the other hand, in the case y m = z m = −1 for all m, i.e. the case essentially equivalent to the ultradiscrete Painlevé VI without parity variables introduced by Ormerod [7] (see also [13]), we have the uniqueness of the initial value problem (see equations (13), (16)).
The ultradiscrete Riccati equation in this paper never appear in the case y m = z m = −1, i.e. the case without parity variable discussed in [7]. Therefore ultradiscretization with parity variable is essential to obtain the ultradiscrete Riccati equation.

Solutions of the ultradiscrete Riccati-type equation
We directly investigate solutions of the ultradiscrete Riccati-type equation with parity variables (see equations (7), (8)), which are also solutions of the ultradiscrete Painlevé VI equation with parity variables (equations (3), (4)). Although it would be natural to investigate solutions of the ultradiscrete Riccati-type equation by introducing the ultradiscrete hypergeometric equation, we leave it to a future problem. In this subsection, we assume that Q > 0, To find solutions of the ultradiscrete Riccati-type equation with parity variables, we set an ansatz that y m = −1 and z m = +1. Then equations (7), (8) are written as We specify the maximum in each term by Then we have (m+1) c is a constant which satisfies inequalities (20). Thus we have four solutions of equations (7), (8) which has a parameter c or c with conditions as follows: condition for equation (7) : condition for equation (8) : condition for equation (7) : condition for equation (8) : condition for equation (7) : condition for equation (8) : condition for equation (7) : condition for equation (8) : We also have solutions of equations (7), (8) which do not contain parameters. One of them is the following solution with the conditions: Besides this type, we have four types of solutions for m 0 in each case of the signs (y m , z m ) = (−1, +1), (+1, −1) and (+1, +1) respectively. We also have four types of solutions for m 0 in each case of the signs (y m , z m ) = (−1, +1), (+1, −1) and (+1, +1) respectively. One of the solutions is as follows: We can construct global solutions of the ultradiscrete Riccati-type equation with parity variables by patching solutions for each region of the variable m suitably. If there exists a value c such that (the conditions 0 ≤ h ≤ Q and 0 ≤ h ≤ Q are implied), we have a solution written as and there exists a value c such that then we have a solution written as then we have a solution written as Note that we have other solutions which can be obtained similarly.

Solutions of the ultradiscrete Painlevé VI equation without parity variables
We now investigate solutions of ultradiscrete Painlevé VI with the fixed parity variable y m = z m = −1 for all m (see equations (13), (16)). We look for the solutions written as Y m = δm+β and Z m = αm+γ for m 0. We substitute them into equations (13), (16). Then we have α + δ = Q, 2(β + γ) + α = B 3 + B 4 + A 1 + A 2 and inequalities among the parameter. More precisely, if then the functions Y m = α m + β , Z m = α m + γ satisfy equations (13), (16). We propose the following conjecture for solutions of ultradiscrete Painlevé VI without parity variables (equations (13), (16)), which is supported by several numerical solutions. On different initial values Y 0 = 43, Z 0 = 50, the solution is In these cases, the conjecture is true.

Concluding remarks
In Section 2, we obtained a ultradiscretization with parity variables of the q-difference Painlevé VI equation. A list of Painlevé-type equations of second order was obtained by Sakai [10], and some members in the list are q-difference Painlevé equations. We believe that ultradiscretization with parity variables of the q-difference Painlevé equations can be done.
Although we investigated solutions of ultradiscrete Riccati-type equation directly, we did not study ultradiscrete hypergeometric equation in this paper. A theory of ultradiscrete hypergeometric equations should be developed because it will have potential for applications to several equations including q-difference hypergeometric equations and ultradiscrete Painlevé equations.
A merit of ultradiscrete equations is that we may have exact solutions with the aid of computers. We formulated Conjecture 1 by calculating several solutions in use of a computer. On the other hand, Murata [5] obtained exact solutions with two parameters for a ultradiscrete Painlevé II equation. We hope to understand exact solutions for ultradiscrete Painlevé equations deeply.