Ultradiscrete Painlevé VI with Parity Variables

We introduce a ultradiscretization with parity variables of the q-difference Painlevé VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of q-Painlevé VI satisfies the ultradiscrete Painlevé VI system of equations with the parity variables, which is valid by using the...

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Datum:	2013
Hauptverfasser:	Takemura, K., Tsutsui, T.
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spelling	irk-123456789-1493622019-02-22T01:23:22Z Ultradiscrete Painlevé VI with Parity Variables Takemura, K. Tsutsui, T. We introduce a ultradiscretization with parity variables of the q-difference Painlevé VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of q-Painlevé VI satisfies the ultradiscrete Painlevé 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 Painlevé VI equation. 2013 Article Ultradiscrete Painlevé VI with Parity Variables / K. Takemura, T. Tsutsui / Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 13 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A13; 34M55; 37B15 DOI: http://dx.doi.org/10.3842/SIGMA.2013.070 http://dspace.nbuv.gov.ua/handle/123456789/149362 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description	We introduce a ultradiscretization with parity variables of the q-difference Painlevé VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of q-Painlevé VI satisfies the ultradiscrete Painlevé 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 Painlevé VI equation.
format	Article
author	Takemura, K. Tsutsui, T.
spellingShingle	Takemura, K. Tsutsui, T. Ultradiscrete Painlevé VI with Parity Variables Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Takemura, K. Tsutsui, T.
author_sort	Takemura, K.
title	Ultradiscrete Painlevé VI with Parity Variables
title_short	Ultradiscrete Painlevé VI with Parity Variables
title_full	Ultradiscrete Painlevé VI with Parity Variables
title_fullStr	Ultradiscrete Painlevé VI with Parity Variables
title_full_unstemmed	Ultradiscrete Painlevé VI with Parity Variables
title_sort	ultradiscrete painlevé vi with parity variables
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149362
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT takemurak ultradiscretepainleveviwithparityvariables AT tsutsuit ultradiscretepainleveviwithparityvariables
first_indexed	2025-07-12T21:56:42Z
last_indexed	2025-07-12T21:56:42Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 070, 12 pages Ultradiscrete Painlevé VI with Parity Variables Kouichi TAKEMURA and Terumitsu TSUTSUI Department of Mathematics, Faculty of Science and Technology, Chuo University, 1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan E-mail: takemura@math.chuo-u.ac.jp Received July 15, 2013, in final form November 11, 2013; Published online November 19, 2013 http://dx.doi.org/10.3842/SIGMA.2013.070 Abstract. We introduce a ultradiscretization with parity variables of the q-difference Painlevé VI system of equations. We show that ultradiscrete limit of Riccati-type solu- tions of q-Painlevé VI satisfies the ultradiscrete Painlevé 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 Painlevé VI equation. Key words: Painlevé equation; ultradiscrete; numerical solutions 2010 Mathematics Subject Classification: 39A13; 34M55; 37B15 1 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]. The ultradiscrete Painlevé equations are systems of cellular automaton and they are obtained by suitable limits (ultradiscretization) from the q-Painlevé equations [11]. We now explain ultradiscretization. We take positive variables x, y, z and set x = exp(X/ε), y = exp(Y/ε) and z = exp(Z/ε). If the variables x, y, z satisfies x + y = z, then we have max(X,Y ) = Z by the limit ε → +0, which follows from the formula lim ε→+0 ε log(exp(X/ε) + exp(Y/ε)) = max(X,Y ). 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) b3b4 = (y(t)− ta1)(y(t)− ta2) (y(t)− a3)(y(t)− a4) , y(t)y(qt) a3a4 = (z(qt)− tb1)(z(qt)− tb2) (z(qt)− b3)(z(qt)− b4) , (1) with the constraint b1b2a3a4 = qa1a2b3b4. The original Painlevé VI equation is recovered by the limit q → 1 (see [3]). mailto:takemura@math.chuo-u.ac.jp http://dx.doi.org/10.3842/SIGMA.2013.070 2 K. Takemura and T. Tsutsui The q-Painlevé VI system has Riccati-type solutions in the special cases [3]. Namely, if b1a3 = qa1b3, b2a4 = a2b4 and the functions y(t) and z(t) satisfy the following Riccati-type equation: z(qt) = b4 y(t)− ta2 y(t)− a4 , y(qt) = a3 z(qt)− tb1 z(qt)− b3 , (2) 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 variab- les. For each value m of the independent variable, we associate the signs ym, zm ∈ {±1} and the amplitudes Ym, Zm ∈ 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 (ym, Ym) and (zm, Zm) by max [ max(A1, A2) +mQ+ Ym +B3 +B4 + S(ym), max(2Ym, A3 +A4) + Zm + Zm+1 + S(zmzm+1), max(A3, A4) + Ym + Zm + Zm+1 + S(−ymzmzm+1) ] = max [ max(2mQ+A1 +A2, 2Ym) +B3 +B4, max(A1, A2) +mQ+ Ym +B3 +B4 + S(−ym), max(2Ym, A3 +A4) + Zm + Zm+1 + S(−zmzm+1), max(A3, A4) + Ym + Zm + Zm+1 + S(ymzmzm+1) ] , (3) max [ max(B1, B2) +mQ+ Zm+1 +A3 +A4 + S(zm+1), max(2Zm+1, B3 +B4) + Ym + Ym+1 + S(ymym+1), max(B3, B4) + Ym + Ym+1 + Zm+1 + S(−ymym+1zm+1) ] = max [ max(2mQ+B1 +B2, 2Zm+1) +A3 +A4, max(B1, B2) +mQ+ Zm+1 +A3 +A4 + S(−zm+1), max(2Zm+1, B3 +B4) + Ym + Ym+1 + S(−ymym+1), max(B3, B4) + Ym + Ym+1 + Zm+1 + S(ymym+1zm+1) ] , (4) with the constraint B1 +B2 +A3 +A4 = Q+A1 +A2 +B3 +B4. (5) 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 ym = zm = −1. Equa- tions (3), (4) are obtained by ultradiscretization of the q-Painlevé VI system under the condition ai > 0 and bi > 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 ultra- discrete 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 ym = zm = −1. We try to study the solutions of the ultradiscrete Riccati-type equation and those of the ultra- discrete Painlevé VI equation. We give examples of solutions which are described by piecewise- linear functions. Based on numerical calculations, we present a conjecture that the solutions Ultradiscrete Painlevé VI with Parity Variables 3 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 Riccati- type 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. 2 Ultradiscretization 2.1 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 t = qm, q = eQ/ε, ai = eAi/ε, bi = eBi/ε, i = 1, 2, 3, 4, y ( qm ) = (s(ym)− s(−ym))eYm/ε, z(qm) = (s(zm)− s(−zm))eZm/ε, (6) and define the parity functions s(ζ) and S(ζ) by s(ζ) = { 1, ζ = +1, 0, ζ = −1, S(ζ) = { 0, ζ = +1, −∞, ζ = −1. Then we have s(ζ) = eS(ζ)/ε for ε > 0. We substitute equation (6) into the equation z(qt)(y(t)− a4) = b4(y(t)− ta2), which is equivalent to first equation of (2), and transpose the terms to disappear the minus signs. Then we have s(zm+1)e Zm+1/εs(ym)eYm/ε + s(−zm+1)e Zm+1/ε { s(−ym)eYm/ε + eA4/ε } + eB4/ε { s(−ym)eYm/ε + e(mQ+A2)/ε } = eB4/εs(ym)eYm/ε + s(zm+1)e Zm+1/ε { s(−ym)eYm/ε + eA4/ε } + s(−zm+1)e Zm+1/εs(ym)eYm/ε. By using the formula s(y)s(z) + s(−y)s(−z) = s(yz) and taking the limit ε→ +0, we have max [ mQ+A2 +B4, Zm+1 +A4 + S(−zm+1), Ym +B4 + S(−ym), Ym + Zm+1 + S(ymzm+1) ] = max [ Zm+1 +A4 + S(zm+1), Ym +B4 + S(ym), Ym + Zm+1 + S(−ymzm+1) ] . (7) It follows from the second equation of (2) that max [ mQ+A3 +B1, Ym+1 +B3 + S(−ym+1), Zm+1 +A3 + S(−zm+1), Ym+1 + Zm+1 + S(ym+1zm+1) ] = max [ Zm+1 +A3 + S(zm+1), Ym+1 +B3 + S(ym+1), Ym+1 + Zm+1 + S(−ym+1zm+1) ] . (8) We call equations (7), (8) the ultradiscrete Riccati-type equation with parity variables. By the ultradiscrete limit, the conditions b1a3 = qa1b3, b2a4 = a2b4 correspond to B1+A3 = Q+A1+B3 and B2 +A4 = A2 +B4. 4 K. Takemura and T. Tsutsui We write the equations for the amplitude variables by fixing the parity variables. Equation (7) for each case is written as zm+1 = 1, ym = 1 ⇒ max(mQ+A2 +B4, Ym + Zm+1) = max(Zm+1 +A4, Ym +B4), (9) zm+1 = 1, ym = −1 ⇒ max(mQ+A2, Ym) +B4 = Zm+1 + max(Ym, A4), zm+1 = −1, ym = 1 ⇒ max(mQ+A2 +B4, Zm+1 +A4) = Ym + max(Zm+1, B4), zm+1 = −1, ym = −1 ⇒ −∞ = max(mQ+A2 +B4, Ym + Zm+1, Zm+1 +A4, Ym +B4). There is no solution in the case zm+1 = ym = −1. Equation (8) for each case is written as zm+1 = 1, ym+1 = 1 ⇒ max(mQ+A3 +B1, Ym+1 + Zm+1) = max(Zm+1 +A3, Ym+1 +B3), (10) zm+1 = 1, ym+1 = −1 ⇒ max(mQ+A3 +B1, Ym+1 +B3) = Zm+1 + max(Ym+1, A3), zm+1 = −1, ym+1 = 1 ⇒ max(mQ+B1, Zm+1) +A3 = Ym+1 + max(Zm+1, B3), zm+1 = −1, ym+1 = −1 ⇒ −∞ = max(mQ+A3 +B1, Ym+1 + Zm+1, Zm+1 +A3, Ym+1 +B3). There is no solution in the case zm+1 = ym+1 = −1. 2.2 Ultradiscretization of the q-Painlevé VI equation We can obtain similarly the ultradiscrete limit with parity variables of the q-Painlevé VI equa- tion. We take a limit of equation (1) as ε→ +0 by setting the values as equation (6). By using the formulae s(y) + s(−y) = 1, s(y)s(−y) = 0, s(y)2 = s(y), s(y)s(z) + s(−y)s(−z) = s(yz), s(y)s(z)s(w) + s(−y)s(−z)s(w) + s(y)s(−z)s(−w) + s(−y)s(z)s(−w) = s(yzw), 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 b1b2a3a4 = qa1a2b3b4 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 ym = 1, zmzm+1 = 1 ⇒ max(max(2Ym, A3 +A4) + Zm + Zm+1,max(A1, A2) +mQ+ Ym +B3 +B4) = max(max(2mQ+A1 +A2, 2Ym) +B3 +B4,max(A3, A4) + Ym + Zm + Zm+1), (11) ym = 1, zmzm+1 = −1 ⇒ max(max(A3, A4) + Ym + Zm + Zm+1,max(A1, A2) +mQ+ Ym +B3 +B4) = max(max(2mQ+A1 +A2, 2Ym) +B3 +B4,max(2Ym, A3 +A4) + Zm + Zm+1), (12) Ultradiscrete Painlevé VI with Parity Variables 5 ym = −1, zmzm+1 = 1 ⇒ Zm + Zm+1 + max(A3, Ym) + max(A4, Ym) = B3 +B4 + max(mQ+A1, Ym) + max(mQ+A2, Ym), (13) ym = −1, zmzm+1 = −1 ⇒ −∞ = max(max(2Ym, A3 +A4) + Zm + Zm+1,max(A1, A2) +mQ+ Ym +B3 +B4, max(2mQ+A1 +A2, 2Ym) +B3 +B4,max(A3, A4) + Ym + Zm + Zm+1)). In the case ym = −1, zmzm+1 = 1, we used the associativity of the maximum and the addition, i.e. max(X1 +W1, X2 +W1, X1 +W2, X2 +W2) = max(X1, X2) + max(W1,W2). (14) There is no solution in the case ym = zmzm+1 = −1. On the other hand, equation (4) for each case is written as zm+1 = 1, ymym+1 = 1 ⇒ max(max(2Zm+1, B3 +B4) + Ym + Ym+1,max(B1, B2) +mQ+ Zm+1 +A3 +A4) = max(max(2mQ+B1 +B2, 2Zm+1) +A3 +A4,max(B3, B4)+Ym+Ym+1+Zm+1), (15) zm+1 = 1, ymym+1 = −1 ⇒ max(max(B3, B4) + Ym + Ym+1 + Zm+1,max(B1, B2) +mQ+A3 +A4 + Zm+1) = max(max(2mQ+B1 +B2, 2Zm+1) +A3 +A4,max(2Zm+1, B3 +B4) + Ym + Ym+1), zm+1 = −1, ymym+1 = 1 ⇒ Ym + Ym+1 + max(B3, Zm+1) + max(B4, Zm+1) = A3 +A4 + max(mQ+B1, Zm+1) + max(mQ+B2, Zm+1), (16) zm+1 = −1, ymym+1 = −1 ⇒ −∞ = max(max(2Zm+1, B3 +B4) + Ym + Ym+1,max(B1, B2) +mQ+ Zm+1 +A3 +A4, max(2mQ+B1 +B2, 2Zm+1) +A3 +A4,max(B3, B4) + Ym + Ym+1 + Zm+1)). There is no solution in the case zm+1 = ymym+1 = −1. We can also obtain the ultradiscrete Painlevé VI system of equations which admits the parities of parameters. Set ai = (s(ai)− s(−ai))eAi/ε, bi = (s(bi)− s(−bi))eBi/ε, i = 1, 2, 3, 4, in addition to equation (6), where ai, bi ∈ {±1} represent the signs of ai, bi (i = 1, 2, 3, 4). We take the ultradiscrete limit (ε→ +0). Then we have max [ 2mQ+A1 +A2 +B3 +B4 + S(−a1a2b3b4), 2Ym +B3 +B4 + S(−b3b4), Ym +mQ+A1 +B3 +B4 + S(a1b3b4ym), Ym +mQ+A2 +B3 +B4 + S(a2b3b4ym), 2Ym + Zm + Zm+1 + S(zmzm+1), Zm + Zm+1 +A3 +A4 + S(a3a4zmzm+1), Ym + Zm + Zm+1 +A3 + S(−a3ymzmzm+1), Ym + Zm + Zm+1 +A4 + S(−a4ymzmzm+1) ] = max [ 2mQ+A1 +A2 +B3 +B4 + S(a1a2b3b4), 2Ym +B3 +B4 + S(b3b4), Ym +mQ+A1 +B3 +B4 + S(−a1b3b4ym), 6 K. Takemura and T. Tsutsui Ym +mQ+A2 +B3 +B4 + S(−a2b3b4ym), 2Ym + Zm + Zm+1 + S(−zmzm+1), Zm + Zm+1 +A3 +A4 + S(−a3a4zmzm+1), Ym + Zm + Zm+1 +A3 + S(a3ymzmzm+1), Ym + Zm + Zm+1 +A4 + S(a4ymzmzm+1) ] , (17) max [ 2mQ+A3 +A4 +B1 +B2 + S(−a3a4b1b2), 2Zm+1 +A3 +A4 + S(−a3a4), Zm+1 +mQ+A3 +A4 +B1 + S(a3a4b1zm+1), Zm+1 +mQ+A3 +A4 +B2 + S(a3a4b2zm+1), 2Zm+1 + Ym + Ym+1 + S(ymym+1), Ym + Ym+1 +B3 +B4 + S(b3b4ymym+1), Ym + Ym+1 + Zm+1 +B3 + S(−b3ymym+1zm+1), Ym + Ym+1 + Zm+1 +B4 + S(−b4ymym+1zm+1) ] = max [ 2mQ+A3 +A4 +B1 +B2 + S(a3a4b1b2), 2Zm+1 +A3 +A4 + S(a3a4), Zm+1 +mQ+A3 +A4 +B1 + S(−a3a4b1zm+1), Zm+1 +mQ+A3 +A4 +B2 + S(−a3a4b2zm+1), 2Zm+1 + Ym + Ym+1 + S(−ymym+1), Ym + Ym+1 +B3 +B4 + S(−b3b4ymym+1), Ym + Ym+1 + Zm+1 +B3 + S(b3ymym+1zm+1), Ym + Ym+1 + Zm+1 +B4 + S(b4ymym+1zm+1) ] , (18) with the constraint B1 +B2 +A3 +A4 = Q+A1 +A2 +B3 +B4, a1a2a3a4 = b1b2b3b4. By setting ai = bi = +1 (i = 1, 2, 3, 4), we recover equations (3), (4). In the rest of the paper, we consider the case ai = bi = +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: Proposition 1. Let no ∈ Z, ỹo, z̃o ∈ {±1} and yo, zo ∈ R. Then there exists a solution (yn, Yn), (zn, Zn) (n ∈ Z) of equations (3), (4) which satisfies the condition (yno , Yno) = (ỹo, yo) and (zno , Zno) = (z̃o, zo). Proof. We show that, if the values (zm, Zm) and (ym, Ym) are fixed, then there exists (zm+1, Zm+1) such that equation (3) is satisfied. If ym = −1, then we have zm+1 = zm and Zm+1 is determined by equation (13). Assume that ym = 1. Write U = max(2Ym, A3 + A4), U ′ = max(A3, A4) + Ym, V = max(2mQ + A1 + A2, 2Ym) and V ′ = max(A1, A2) + mQ + Ym. If U ≥ U ′ and V ≥ V ′ (resp. U < U ′ and V < V ′), then equation (3) is satisfied by setting zm+1 = zm and Zm+1 = V −U+B3+B4−Zm (resp. Zm+1 = V ′−U ′+B3+B4−Zm) (see equation (11)). If U ≥ U ′ and V < V ′ (resp. U < U ′ and V ≥ V ′), then equation (3) is satisfied by setting zm+1 = −zm and Zm+1 = V ′−U+B3+B4−Zm (resp. Zm+1 = V −U ′+B3+B4−Zm) (see equation (12)). Similarly there exists (ym+1, Ym+1) such that equation (4) is satisfied while (ym, Ym) and (zm+1, Zm+1) are fixed. We proceed it for m = no, no + 1, . . . , and a solution (yn, Yn), (zn, Zn) (n > no) is obtained. By applying a similar procedure to equation (4) for n = no − 1, equation (3) for n = no − 1 Ultradiscrete Painlevé VI with Parity Variables 7 and so on, a solution (yn, Yn), (zn, Zn) (n < no) is obtained. Thus we have a solution (Yn, Zn) (n ∈ Z) which satisfies the condition (yno , Yno) = (ỹo, yo) and (zno , Zno) = (z̃o, zo). � 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 Ym = A3, A4, A1 + mQ or A2 + mQ, we do not have uniqueness of the solution. Similarly in the case Zm+1 = B3, B4, B1 + mQ or B2 + mQ, we do not have uniqueness of the solution. On the other hand, in the case ym = zm = −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)). 3 Ultradiscrete Riccati-type equation and ultradiscrete Painlevé VI equation We show that the ultradiscrete Riccati-type equation with parity variables (equations (7), (8)) satisfies the ultradiscrete Painlevé VI equation with parity variables (equations (3), (4)). Lemma 1. If max(X1, X2) = max(X3, X4) and max(W1,W2) = max(W3,W4), then max(X1 +W1, X3 +W3, X2 +W4, X4 +W2) = max(X2 +W2, X4 +W4, X1 +W3, X3 +W1). (19) This lemma can be shown by case-by-case analysis (see [13], e.g. if X1 ≥ X2, X3 ≥ X4, W1 ≤ W2 and W3 ≤ W4, then X1 = X3, W2 = W4 and l.h.s. of equation (19) = max(X1 + W1, X1 + W3, X2 + W2, X4 + W2) = r.h.s. of equation (19)). Note that it is an ultradiscrete analogue of the identity eX1 − eX3 = eX4 − eX2 , eW1 − eW3 = eW4 − eW2 =⇒ eX1+W1 + eX3+W3 + eX2+W4 + eX4+W2 = eX2+W2 + eX4+W4 + eX2+W3 + eX3+W1 . Theorem 1. If A1 +B3 +Q = A3 +B1 and A2 +B4 = A4 +B2, then any solutions of the ultra- discrete Riccati-type equation with parity variables (equations (7), (8)) satisfy the ultradiscrete Painlevé VI equation with parity variables (equations (3), (4)). Proof. This theorem is shown by the lemma and case-by-case analysis. We show equation (4) for every values of parity variables zm+1, ym, ym+1 by applying equa- tions (7), (8) and the lemma. In the case zm+1 = ym = ym+1 = 1, we set X1 = mQ + A2 + B4, X2 = Ym + Zm+1, X3 = A4 +Zm+1, X4 = B4 + Ym, W1 = mQ+A3 +B1, W2 = Ym+1 +Zm+1, W3 = A3 +Zm+1, W4 = B3 + Ym+1. Then equations (7), (8) are written as max(X1, X2) = max(X3, X4) and max(W1,W2) = max(W3,W4), i.e. equations (9), (10). By applying the lemma, we have max [ (mQ+A2 +B4) + (mQ+A3 +B1), (A4 + Zm+1) + (A3 + Zm+1), (Ym + Zm+1) + (B3 + Ym+1), (B4 + Ym) + (Ym+1 + Zm+1) ] = max [ (Ym + Zm+1) + (Ym+1 + Zm+1), (B4 + Ym) + (B3 + Ym+1), (mQ+A2 +B4) + (A3 + Zm+1), (A4 + Zm+1) + (mQ+A3 +B1) ] , which is equivalent to equation (4) in the case zm+1 = ym = ym+1 = 1 (i.e. equation (15)) by using the relation A2 +B4 = A4 +B2. We can show similarly equation (4) in the cases (zm+1, ym, ym+1) = (1, 1,−1), (1,−1, 1), (1,−1,−1) by applying the lemma. In the case (zm+1, ym, ym+1) = (−1, 1, 1), equation (4) is shown by using equation (14). 8 K. Takemura and T. Tsutsui In the cases (zm+1, ym, ym+1) = (−1,−1, 1), (−1, 1,−1), (−1,−1,−1), there is no solution to equations (7), (8), and the theorem holds true. Similarly equation (3) is shown for every values of parity variables ym+1, zm, zm+1 by applying equations (7), (8) and using the relation A1 +B3 +Q = A3 +B1. For details see [13]. � The ultradiscrete Riccati equation in this paper never appear in the case ym = zm = −1, i.e. the case without parity variable discussed in [7]. Therefore ultradiscretization with parity variable is essential to obtain the ultradiscrete Riccati equation. 4 Solutions 4.1 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, A1 +B3 +Q = A3 +B1 and A2 +B4 = A4 +B2. Set h = A3 +B1 −A2 −B4, h′ = A3 −A4 −B3 +B4. To find solutions of the ultradiscrete Riccati-type equation with parity variables, we set an ansatz that ym = −1 and zm = +1. Then equations (7), (8) are written as max(mQ+A2, Ym) +B4 = Zm+1 + max(A4, Ym), max((m+ 1)Q+A1, Ym+1) +B3 = Zm+1 + max(A3, Ym+1). We specify the maximum in each term by mQ+A2 ≥ Ym, A4 ≤ Ym, (m+ 1)Q+A1 ≥ Ym+1, A3 ≤ Ym+1. (20) Then we have (m+1)Q+A1+B3 = Zm+1+Ym+1, mQ+A2+B4 = Zm+1+Ym and Ym+1−Ym = A3 + B1 − A2 − B4 = h. Hence we obtain solutions Ym = hm + c, Zm+1 = (Q − h)m + A2 + B4 − c where 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: (ym, Ym) = (−1, hm+ c), (zm+1, Zm+1) = (+1, (Q− h)m+A2 +B4 − c), condition for equation (7) : hm ≥ A4 − c, (Q− h)m ≥ c−A2, condition for equation (8) : h(m+ 1) ≥ A3 − c, (Q− h)(m+ 1) ≥ c−A1, (ym, Ym) = (+1, hm+A2 +B4 − c), (zm+1, Zm+1) = (−1, (Q− h)m+ c), condition for equation (7) : (Q− h)m ≥ B4 − c, hm ≥ c−B2, condition for equation (8) : (Q− h)m ≥ B3 − c, hm ≥ c−B1, (ym, Ym) = (−1, h′m+ c′), (zm+1, Zm+1) = (+1, h′m+ c′ +B4 −A4), condition for equation (7) : h′m ≤ A4 − c′, (Q− h′)m ≤ c′ −A2, condition for equation (8) : h′(m+ 1) ≤ A3 − c′, (Q− h′)(m+ 1) ≤ c′ −A1, (ym, Ym) = (+1, h′m+ c′ −B4 +A4), (zm+1, Zm+1) = (−1, h′m+ c′), condition for equation (7) : h′m ≤ B4 − c′, (Q− h′)m ≤ c′ −B2, condition for equation (8) : h′m ≤ B3 − c′, (Q− h′)m ≤ c′ −B1. Ultradiscrete Painlevé VI with Parity Variables 9 We also have solutions of equations (7), (8) which do not contain parameters. One of them is the following solution with the conditions: (ym+1, Ym+1) = (−1, A2 +B4 −B1), (zm+1, Zm+1) = (+1,mQ+B1), A2 +B4 ≤ A3 +B1, B1 ≤ B2, mQ ≥ max(A2 +B4 −A1 −B1, B4 −B1). Besides this type, we have four types of solutions for m� 0 in each case of the signs (ym, zm) = (−1,+1), (+1,−1) and (+1,+1) respectively. We also have four types of solutions for m � 0 in each case of the signs (ym, zm) = (−1,+1), (+1,−1) and (+1,+1) respectively. One of the solutions is as follows: (ym+1, Ym+1) = (+1, A3), (zm+1, Zm+1) = (+1, B4), A3 ≥ A4, B3 ≤ B4, mQ ≤ min(A3 −A2, B4 −B1). 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 max(A1, A4, A2+B4−B1, A1−B1+B2) ≤ c ≤ min(A2, A3, A3−B3+B4, A2+B4−B3), (the conditions 0 ≤ h ≤ Q and 0 ≤ h′ ≤ Q are implied), we have a solution written as (ym, Ym) = { (−1, h′m+ c), m ≤ 0, (−1, hm+ c), m ≥ 1, (zm, Zm) = { (+1, h′m+B3 −A3 + c), m ≤ 0, (+1, (Q− h)m+A1 +B3 − c), m ≥ 1. (21) Let m0 < 0. If 0 ≤ h ≤ Q, 0 ≤ h′ ≤ Q, B3 ≤ B4 ≤ B1 ≤ B4 + Q, A3 + B1 ≥ A4 + B4, max(A2, A4) ≤ A3 and there exists a value c′ such that A4 ≤ h′m0 + c′ ≤ min(A3, A4 + h′), (Q− h′)m0 + max(A1, A2) ≤ c′, then we have a solution written as (ym, Ym) =  (−1, h′m+ c′), m ≤ m0, (+1, A3), m0 + 1 ≤ m ≤ 0, (−1, hm+A2), m ≥ 1, (zm, Zm) =  (+1, h′m+B3 −A3 + c′), m ≤ m0, (+1, B4), m0 + 1 ≤ m ≤ 1, (+1, (Q− h)(m− 1) +B4), m ≥ 2. (22) Let m0 > 0. If 0 ≤ h ≤ Q, 0 ≤ h′ ≤ Q, B4 ≤ B1 ≤ B2, A1 + B1 ≤ A2 + B4 ≤ A1 + B1 + Q, A2 ≤ min(A1, A3) +Q, A1 ≥ A4, A2 +B3 ≤ B4 +A3 +Q and there exists a value c such that h(m0 − 1) +B1 ≤ c ≤ hm0 + min(B1, B2), (Q− h)m0 + c ≥ max(B3 + (Q− h), B4), then we have a solution written as (ym, Ym) =  (+1, h′m+A1 −B1 +B2), m ≤ −1, (−1, A2 +B4 −B1), 0 ≤ m ≤ m0 − 1, (+1, hm+A2 +B4 − c), m ≥ m0, 10 K. Takemura and T. Tsutsui (zm, Zm) =  (−1, h′m+B2 −Q), m ≤ −1, (+1, A2 +B4 −A1 −Q), m = 0, (+1, (m− 1)Q+B1), 1 ≤ m ≤ m0, (−1, (Q− h)(m− 1) + c), m ≥ m0 + 1. (23) Note that we have other solutions which can be obtained similarly. We now fix the parameters by A1 = 25, A2 = 46, A3 = 67, A4 = 23, B1 = 59, B2 = 65, B3 = 1, B4 = 42, Q = 100. Then B1+A3 = A1+B3+Q, A4+B2 = A2+B4, h = 38 and h′ = 85. Equation (21) is a solution of the ultradiscrete Riccati-type equation with parity variables, if 31 ≤ c ≤ 46. Moreover equation (22) (resp. equation (23)) is a solution of the ultradiscrete Riccati-type equation with parity variables, if m0 < 0 and −85m0+23 ≤ c′ ≤ −85m0+67 (resp. m0 > 0 and 21 + 38m0 ≤ c ≤ 59 + 38m0). We also have other solutions. For example, if m0 < 0 and −85m0 − 18 ≤ c′ ≤ −85m0 + 23, then we have the following solution: (ym, Ym) =  (−1, 85m+ c′), m ≤ m0, (+1, 67), m0 + 1 ≤ m ≤ 0, (−1, 38m+ 46), m ≥ 1, (zm, Zm) =  (+1, 85m− 66 + c′), m ≤ m0 + 1, (+1, 42), m0 + 2 ≤ m ≤ 1, (+1, 62m− 20), m ≥ 2. If m0 < 0 and 15m0 + 31 ≤ c′ ≤ 15m0 + 46, then we have the following solution: (ym, Ym) =  (−1, 85m+ c′), m ≤ m0, (−1, 100m+ 31), m0 + 1 ≤ m ≤ 0, (−1, 38m+ 31), m ≥ 1, (zm, Zm) =  (+1, 85m− 66 + c′), m ≤ m0, (+1, 100m− 35), m0 + 1 ≤ m ≤ 0, (+1, 62m− 5), m ≥ 1. 4.2 Solutions of the ultradiscrete Painlevé VI equation without parity variables We now investigate solutions of ultradiscrete Painlevé VI with the fixed parity variable ym = zm = −1 for all m (see equations (13), (16)). We look for the solutions written as Ym = δm+β and Zm = αm+γ for m� 0. We substitute them into equations (13), (16). Then we have α + δ = Q, 2(β + γ) + α = B3 + B4 + A1 + A2 and inequalities among the parameter. More precisely, if 0 ≤ α ≤ Q, 2(β + γ) + α = B3 +B4 +A1 +A2, α(m+ 1) + γ ≥ max(B3, B4), αm+ min(A1, A2) ≥ β, (Q− α)m+ β ≥ max(A3, A4), (Q− α)m+ min(B1, B2) ≥ α+ γ, (24) then the functions Ym = (Q− α)m+ β, Zm = αm+ γ satisfy equations (13), (16). Note that the constraint condition B3 +B4 +A1 +A2 = Q+A3 + A4 +B1 +B2 is used to prove the equalities. Similarly, if 0 ≤ α′ ≤ Q, α′ + 2(γ′ − β′) = B3 +B4 −A3 −A4, Ultradiscrete Painlevé VI with Parity Variables 11 α′(m+ 1) + γ′ ≤ min(B3, B4), α′m+ β′ ≤ min(A3, A4), (Q− α′)m+ max(B1, B2) ≤ α′ + γ′, (Q− α′)m+ max(A1, A2) ≤ β′, (25) then the functions Ym = α′m+ β′, Zm = 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 nu- merical solutions. Conjecture 1. For every solution Ym, Zm (m ∈ Z) of ultradiscrete Painlevé VI without parity variable (equations (13), (16)), there exist m0,m ′ 0 ∈ Z and α, β, γ, α′, β′, γ′ ∈ R satisfying equations (24), (25) such that Ym = α′m+ β′, Zm = α′m+ γ′, m ≤ m′ 0, Ym = (Q− α)m+ β, Zm = αm+ γ, m ≥ m0. We give examples of solutions. We consider solutions in the case A1 = 32, A2 = 33, A3 = 37, A4 = 22, B1 = 53, B2 = 65, B3 = 8, B4 = 4, and Q = 100. We choose the initial values by Y0 = 43, Z0 = 40. Then the solution is written as Ym =  95m+ 111, m ≤ −1, 43, m = 0, 11m+ 111, m ≥ 1, Zm = { 95m+ 40, m ≤ 0, 89m− 117, m ≥ 1. On different initial values Y0 = 43, Z0 = 50, the solution is Ym =  85m− 81, m ≤ −8, −669, m = −7, 115m+ 131, −1 ≤ m ≤ −6, 43, m = 0, −9m+ 131, 1 ≤ m ≤ 11, 9m− 72, m ≥ 12, Zm =  85m− 147, m ≤ −7, 115m+ 50, 0 ≤ m ≤ −6, 109m− 147, 1 ≤ m ≤ 11, 1156, m = 12, 91m+ 65, m ≥ 13. In these cases, the conjecture is true. 5 Concluding remarks In Section 2, we obtained a ultradiscretization with parity variables of the q-difference Painle- vé 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 hypergeo- metric 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 com- puters. 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. 12 K. Takemura and T. Tsutsui Acknowledgments The authors would like to thank Professor Junkichi Satsuma for discussions and suggestions. They also thank the referees for valuable comments. The first author is partially supported by the Grant-in-Aid for Young Scientists (B) (No. 22740107) from the Japan Society for the Promotion of Science. References [1] Bruno A.D., Batkhin A.B. (Editors), Proceedings of the International Conference “Painlevé equations and related topics” (June, 2011, Saint Petersburg, Russia), De Gruyter Proceedings in Mathematics, De Gruyter, Berlin, 2012. [2] Isojima S., Satsuma J., A class of special solutions for the ultradiscrete Painlevé II equation, SIGMA 7 (2011), 074, 9 pages, arXiv:1107.4416. [3] Jimbo M., Sakai H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145–154, chao-dyn/9507010. [4] Mimura N., Isojima S., Murata M., Satsuma J., Singularity confinement test for ultradiscrete equations with parity variables, J. Phys. A: Math. Theor. 42 (2009), 315206, 7 pages. [5] Murata M., Exact solutions with two parameters for an ultradiscrete Painlevé equation of type A (1) 6 , SIGMA 7 (2011), 059, 15 pages, arXiv:1106.3384. [6] Ohta Y., Ramani A., Grammaticos B., An affine Weyl group approach to the eight-parameter discrete Painlevé equation, J. Phys. A: Math. Gen. 34 (2001), 10523–10532. [7] Ormerod C.M., Reductions of lattice mKdV to q-PVI, Phys. Lett. A 376 (2012), 2855–2859, arXiv:1112.2419. [8] Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829–1832. [9] Sakai H., Casorati determinant solutions for the q-difference sixth Painlevé equation, Nonlinearity 11 (1998), 823–833. [10] Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165–229. [11] Takahashi D., Tokihiro T., Grammaticos B., Ohta Y., Ramani A., Constructing solutions to the ultradiscrete Painlevé equations, J. Phys. A: Math. Gen. 30 (1997), 7953–7966. [12] Tokihiro T., Takahashi D., Matsukidaira J., Satsuma J., From soliton equations to integrable cellular auto- mata through a limiting procedure, Phys. Rev. Lett. 76 (1996), 3247–3250. [13] Tsutsui T., Ultradiscretization with parity variables of q-Painlevé VI, Master’s Thesis, Chuo University, 2013 (in Japanese). http://dx.doi.org/10.3842/SIGMA.2011.074 http://arxiv.org/abs/1107.4416 http://dx.doi.org/10.1007/BF00398316 http://arxiv.org/abs/chao-dyn/9507010 http://dx.doi.org/10.1088/1751-8113/42/31/315206 http://dx.doi.org/10.3842/SIGMA.2011.059 http://arxiv.org/abs/1106.3384 http://dx.doi.org/10.1088/0305-4470/34/48/316 http://dx.doi.org/10.1016/j.physleta.2012.09.008 http://arxiv.org/abs/1112.2419 http://dx.doi.org/10.1103/PhysRevLett.67.1829 http://dx.doi.org/10.1088/0951-7715/11/4/004 http://dx.doi.org/10.1007/s002200100446 http://dx.doi.org/10.1088/0305-4470/30/22/029 http://dx.doi.org/10.1103/PhysRevLett.76.3247 1 Introduction 2 Ultradiscretization 2.1 Ultradiscretization of the Riccati-type equation 2.2 Ultradiscretization of the q-Painlevé VI equation 3 Ultradiscrete Riccati-type equation and ultradiscrete Painlevé VI equation 4 Solutions 4.1 Solutions of the ultradiscrete Riccati-type equation 4.2 Solutions of the ultradiscrete Painlevé VI equation without parity variables 5 Concluding remarks References

Ultradiscrete Painlevé VI with Parity Variables

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