Symmetry, Integrability and Geometry: Methods and Applications A Class of Special Solutions for the Ultradiscrete Painlevé II Equation ⋆

A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.


Introduction
Ultradiscretization [1] is a limiting procedure transforming a given difference equation into a cellular automaton, in which dependent variables also take discrete values. To apply this procedure, we first replace a dependent variable x n in the equation by where ε is a positive parameter. Next, we apply ε log to both sides of the equation and take the limit ε → +0. Then, using identity lim ε→+0 ε log(e X/ε + e Y /ε ) = max(X, Y ), the original difference equation is approximated by a piecewise linear equation which can be regarded as a time evolution rule for a cellular automaton. In many examples, cellular automata obtained by this systematic method preserve the essential properties of the original equations, such as the qualitative behavior of exact solutions. However, the ansatz (1) is only possible if the variable x n is positive definite. This restriction is called 'negative problem'. From theoretical and application points of view, it is an interesting problem to study ultradiscrete analogs of special functions and their defining equations, including the Painlevé equations. Ultradiscrete analogs for some of the Painlevé equations are proposed, for example, in [2,3,4]. However, the class of ultradiscretizable Painlevé equations has been restricted because of the negative problem. Some attempts resolving this problem are reported, for example, in [5,6,7]. The authors and coworkers study in [5] an ultradiscrete Painlevé II equation with sinh ansatz and discuss its special solution of Bi function type.
In order to overcome the negative problem, a new method 'ultradiscretization with parity variables' (p-ultradiscretization) is proposed in [8]. The procedure keeps track of the sign of original variables. By using this method, the authors and coworkers present [9] a p-ultradiscrete analog of the q-Painlevé II equation (q-PII), (z(qτ )z(τ ) + 1) z(τ )z(q −1 τ ) + 1 = aτ 2 z(τ ) τ − z(τ ) . (2) In [9], we also discuss a series of special solutions corresponding to that of q-PII written in the determinants of size N . However, the resulting solutions are reduced to only one solution for the p-ultradiscrete Painlevé II (udPII) equation. In this paper, we construct other series of special solutions for udPII and discuss their structure. In section 2, we introduce the results in [9] for the p-ultradiscrete Airy equation. Then, we construct special solutions for udPII in section 3. Finally, concluding remarks are given in section 4.

Ultradiscrete Airy equation with parity variables
We start with a q-difference analog of the Airy equation which reduces to the Airy equation d 2 v ds 2 + sv = 0 in a continuous limit.
Taking the ultradicrete limit, we obtain a p-ultradiscrete analog of the Airy equation where S(ω) is defined by An ultradiscretized variable is represented by a pair of ω m and W m , which is denoted as W n = (ω m , W m ) in what follows. It is possible to rewrite the implicit form (4) into explicit forward schemes Note that we generally have both of unique and indeterminate schemes depending on given values of (ω m , W m ) and (ω m−1 , W m−1 ). The explicit backward schemes are obtained by replacing m ± 1 with m ∓ 1, respectively. We find two typical solutions of (4). One is an Ai-function-type solution for W 0 = (+1, 0) and W 1 = (+1, 0), and the other is a Bi-function-type for W 0 = (+1, 0) and W 1 = (−1, 0), They show similar behavior as those of the Ai and Bi functions, respectively.

Ultradiscrete Painlevé II equation with parity variables
For the following discussion, we first introduce the results for (2). It has been shown in [10] that solves (2) with a = q 2N +1 , where the functions g (N ) (t) (N ∈ Z) satisfy the bilinear equations for N ≥ 0 and for N < 0. It is also known that g (N ) (τ ) are written in terms of the Casorati determinant of size |N | whose elements are represented by the solutions of (3). In order to construct ultradiscrete analogs of these equations, we put τ = q m , q = e Q/ε (Q < 0) and a = e A/ε . Furthermore, we introduce Then (2) is reduced to udPII, For (6) and (7), we have their ultradiscrete analogs and max 2NQ + G respectively. For (8) and (9), we have and max 2(N + 1)Q + G respectively. Finally, the transformations (5) are reduced to for N ≥ 0 and for N < 0. If we find solutions for the ultradiscrete bilinear equations, special solutions for udPII are obtained through (15) -(18).
Hereafter we consider only the case of A = (2N +1)Q in (10), which corresponds to a = q 2N +1 in the discrete system. Firstly, we present the results reported in [9], that is, the Ai-functiontype solutions for N ≥ 0. Solutions of (13) and (14) are given by G m → −∞ as m → −∞ in the same way as the uAi function, we call these solutions the Ai-function-type solutions. From these solutions, we have only one special solution of udPII with A = (2N + 1)Q for N = 0, 1, 2, . . .
which does not depend on N . We note that Z (11) and (12) with N = 0. By using this reslut, we inductively construct solutions G . We further assume that G for any m are uniquely determined in (11) and (12). Then we have the following solutions G m → ∞ as m → −∞ in the same way as the uBi function, we call these solutions the Bi-function-type solutions.
By substituting these solutions into (15) and (16), we obtain special solutions of udPII, We notice that (19) and (20) have different asymptotic behavior in Z (N ) m for m → −∞ and in the phases for m > 0. Furthermore, we remark that (20) have N -dependence.
Thirdly, we study Ai-function-type solutions for N < 0. We find that G  (13) and (14) with N = −1. Starting from these simple solutions, we inductively find the solutions G (17) and (18), we have special solutions of udPII with A = (2N + 1)Q for N = −1, −2, . . . , Typical behavior of these solutions is shown in Figure 1. They converge to 0 as m → −∞ and oscillate for m ≥ −N . It is interesting to note that (21) constructed from the uAi function has essentially the same structure as (20) constructed from the uBi function. Finally, we study Bi-function-type solutions for N < 0. We find that G (13) and (14) (13) and (14). We again inductively obtain the solutions G From these solutions, we have special solutions of udPII, Typical behavior of these solutions is shown in Figure 2. Note that (21) and (22)

Concluding Remarks
In this paper we have presented a class of special solutions for the p-ultradiscrete analog of q-PII. The solutions are classified into four groups; Ai-function-type and Bi-function-type solutions for the system parameter N ≥ 0, and those for N < 0. In the preceding paper [9] are given only the Ai-function-type solutions for N ≥ 0, which do not depend on N . Three other groups which are newly given in this paper do depend on N . Moreover, the solutions of each group have different structures. For example, we observe differences between the Ai-and Bi-function-type solutions in their asymptotic amplitude and phases, which may reflect the structure of solutions of difference and continuous equations. The Bi-function-type solutions for N ≥ 0 have fairly complicated internal structure, although we do not know the origin of these structures yet. At any rate, these results may indicate the richness of solution space of the ultradiscrete equation.
For the continuous and discrete Airy equations, linear combination of Ai and Bi functions give their general solutions. In the ultradiscrete case, max(f, g) corresponds to the linear combination of functions f and g. Hence, we believe that the cases we treated in this paper cover quite wide class of special solutions of the ultradiscrete equations.
Our method of constructing solutions is intuitive and purely based on the ultradiscrete equations. We believe that the solutions we obtain correspond to those of q-PII represented by the Casorati determinant of size |N | whose elements are given by the q-difference Ai or Bi function. It is a future problem to clarify the relationship between discrete and ultradiscrete solutions through a limiting procedure. It is also a future problem to construct p-ultradiscrete analogs of other Painlevé equations and their special solutions. Ultradiscrete singularity confinement test, which is an integrability detector for ultradiscrete equations with parity variables, is applied to various ultradisctete equations.
The ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts. Exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures.

I. ULTRADISCRETISATION WITH PARITY VARIABLES
In order to show how ultradiscretisation with parity variables works in a simple case, we choose the discrete Riccati equation Using the discrete Cole-Hopf transformation the mapping (1) can be transformed into the second order linear difference equation The gerenal solution to (3), neglecting the case a = b, is given by Substituting (4) back into (2) gives the following general solution x n = a n + cb n a n−1 + cb n−1 , where c = c 2 /c 1 . Note that x n → max(a, b) as n → ∞.
We now turn our attention to ultradiscretising (1). For brevity, we shall assume a, b > 0. The first step in ultradiscretisation of (1) with parity variables is to introduce a new dependent variable σ n ∈ {−1, 1} (the parity variable), and write x n = σ nxn wherex n > 0 is another dependent variable (the amplitude variable which is always positive). We further introduce a function s defined by and write σ n = s(σ n ) − s(−σ n ). Once we collect non-negative terms to each side of equality and use the exponential ansatz a = e A/δ , b = e B/δ andx = e X/δ , we obtain the following implicit equation for σ n and X n in the limit δ → +0: 2 where the function S is defined by Equation (7) can be rewritten as the following pair of equations: and X n+1 where the signum function "sgn" is defined by We will refer to (9) and (10) as the ultradiscrete Riccati equations. In what follows, we use the notation X n := (σ n , X n ) to denote the pair of dependent variables σ n and X n .
Given the initial values U 0 = (τ 0 , U 0 ) and U 1 = (τ 1 , U 1 ), the solutions to the linearised ultradiscrete Riccati equations (14) and (15) are as follows, where for brevity, we consider only the case A = B. and 2. Given some integer k(≥ 1), if τ n = τ 0 and U n = n{A 3. Otherwise, and In all cases, the parity variable τ tends to a constant while the amplitude variable U either grows linearly or tends to a constant, depending on the signs of A and B. (12) and (13),

Substituting (16)-(21) back into the ultradiscrete Cole-Hopf transformation
we obtain the following solution to the ultradiscrete Riccati equations (9) and (10): and 2. Given some integer k(≥ 1), if σ n = 1 and 3. Otherwise, we have and In all cases, we see that the solution tends to a constant value. This is analogous to the long time behaviour of the solutions of continuous and discrete Riccati equations.

ULTRADISCRETE SYSTEMS
The singularity confinement criterion for ultradiscrete systems (with parity variables) was introduced by some of the present authors as an integrability detector i.e. as a criterion which would allow one to distinguish between integrable and nonintegrable systems. The pertinence of this criterion was displayed prominently 5 where we have detailed its workings.
As an illustrative example, let us consider the autonomous limit of multiplicative dP I 12 x n+1 x n x n−1 = ax n + 1.
This equation has an invariant and is integrable. A singular point of a second order difference equation is defined as the particular point x n such that ∂x n+1 /∂x n−1 = 0 and ∂x n−k /∂x n−1 = 0 (∀k ≥ 1) hold for generic x n−1 . The singularity pattern is defined as the set of values {x n , x n+1 , x n+2 , . . .} such that ∂x n+k /∂x n−1 = 0 (k ≥ 1) hold. By this definition, the (unique) singular point of (28) is x n = −1/a and its confined singularity pattern is given by {−1/a, 0, ∞, ∞, 0, −1/a} where regular values extend outside of this sequence.
Equation (28) can be ultradiscretised with parity variables for a > 0 to the following pair of equations: and In what follows, we assume A > 0. The singular point of a second order ultradiscrete equation with parity variables is defined as the particular point X n such that X n+1 is not uniquely determined for generic initial value X n−1 5 . The singularity pattern is defined as the set of values {X n+1 , X n+2 , . . .} that depend on the indeterminacy X n+1 . By this definition, the singular point of (29) and (30) is X n = (−1, −A), and taking this singular point gives rise to the singularity X n+1 = (σ n+1 , X n+1 ) where σ n+1 = indeterminate and X n+1 ≤ A − X n−1 .
Just as in the discrete case, to test (29) and (30) for ultradiscrete singularity confinement we iterate with an initial value X 0 = (σ 0 , X 0 ) and the singular point X 1 = (−1, −A), and see how the singularity propagates. We see that the range X 2 ≤ A − X 0 depends on the initial value X 0 , and consequently the subsequent evolution depend on X 0 and how we choose the quantity X 2 . For the case X 0 < −A, there exist the following three distinct singularity patterns.
3. If we choose 2A < X 2 ≤ A − X 0 , then In (31), we see that the reappearance of the singular point at X 6 gives rise to a new indeterminate value X 7 . The subsequent values are independent of the singularity X 2 , and we call such a singularity pattern confined. Similarly, the singularity in (32) is also confined.
In (33), on the other hand, the dependence on X 2 propagates indefinitely, and we call such a singularity pattern unconfined.
For other initial values X 0 we obtain only confined patterns regardless of the choice of X 2 , namely (31) and (32) for the case −A < X 0 < 2A, and (31) for the case X 0 > 2A.
The integrability criterion 5 is applied to our example in the following way. For X 0 > −A, the singularity is confined. For X 0 < −A, there exist both confined and unconfined singularity patterns, but the singularity can be made confined provided we restrict the range of the singularity X 2 ≤ A − X 0 to X 2 < 2A. In short, for any given initial value X 0 , the singularity is confined with a restriction on the singularity X 2 whenever necessary. In general, whenever, perhaps with the appropriate constraint which force a singularity to confine, we have only confined singularities for any initial value of X 0 , we consider that our integrability criterion is satisfied. Therefore we conclude that (29) and (30) satisfy the integrability criterion.
Note that the singular point x n = 0 of (34) vanishes in taking an ultradiscrete limit. We therefore use a change of variablex n = x n + 1 to shift this singular point tox n = 1, and ultradiscretize the resulting equation forx with parity variables. This leads to the following pair of equations: and = arbitrary if σ n = 1 and X n = 0 ≤ 0 if σ n = −1 and X n = 0 ≥ X n−1 ≤ X n if σ n = σ n−1 and X n = X n−1 > 0 ≤ 0 if σ n−1 = 1 and X n−1 = 0 > X n = max(X n , X n−1 , 0) otherwise.
In (37)-(39) the singularity X 2 is confined, but in (40) and (43) the singularity propagates indefinitely. For any X 0 , the singularity is confined provided we impose the constraint X 2 < 0, and therefore the ultradiscrete Hietarinta-Viallet equations satisfy the integrability criterion.
However, one remark needs to be addressed here. That is, the confined singularity patterns (37)-(39) do not correspond to the one obtained from the difference equation (34). The ultradiscrete equivalent of the singularity pattern of (34) is (41). It is interesting to note that the ultradiscrete Hietarinta-Viallet equations satisfy the integrability criterion but its confined singularity structure is not the one found from the difference equation. x n+1 = ax n−1 x n + a x n + 1 .
Equation (44)  Since there exists one singular point whose singularity pattern is unconfined, the integrability criterion is violated.
In what follows, we assume A > 0. The singularity pattern through the first singular point X 1 = (−1, 0) under the condition X 0 > −A is given below: The singularity in (47) is confined. For X 0 < −A, there also exist unconfined patterns but the confined pattern (47) is obtained provided we choose X 2 > A. Thus, as far as this first singular point is concerned, the singularity is confined.
On the other hand, the singularity pattern through the second singular point X 1 = (−1, A) with X 0 < −2A is as follows.
X 2 = (σ 2 , X 2 ), where σ 2 = indeterminate and X 2 ≤ A + X 0 (48) As expected from the discrete case, the singularity in (48) is not confined. Since there exists one case for the initial value X 0 such that no confined singularity pattern exists, (45) and (46) do not satisfy the integrability criterion.
The general expression for the pattern (47) can be found from the exact solutions of the ultradiscrete equations (45), (46).

V. CONCLUSION
In this paper, we have investigated the integrability detector for ultradiscrete systems with parity variables 5 . This detector is a transposition of the singularity cofinement discrete integrability detector to an ultradiscrete setting. The singularity confinement criterion is based on the requirement that every spontaneously appearing singularity disappears after a certain number of iteration steps. In the case of ultradiscrete systems with parity variables under consideration, the situation is somewhat more complex. If for all initial conditions only confined singularities exist, then the system does satisfy the confinement criterion, while if for some initial condition there exist only unconfined singularities, then the confinement criterion is not satisfied. The situation is more subtle when for some initial condition there exist both confined and unconfined singularity patterns. The way to apply our criterion in this case is to examine the unconfined singularities and the conditions for their existence: 15 if with the application of suitable constraints that are independent of the initial condition we can make the unconfined singularities disappear, then we claim that the singularity confinement criterion is satisfied.
One advantage of this integrability detector is that it is applicable to a wider class of ultradiscrete equations, including the ones obtained from difference equations of the form x n+1 +x n−1 = f (x n ). The Hietarinta-Viallet equation (34) is such an example. The criterion works in close parallel to that of the discrete case, and consequently there exist nonintegrable ultradiscrete equations the singularities of which are confined. At that level, the results based on our method are in perfect parallel to the ones obtained with the Joshi-Larfortune approach (but we must stress once more that our method has a much wider applicability).
The application of our criterion to linearisable systems led mostly to results which were expected but also to, on the surface, a puzzling one. Just as in the discrete case, the projective systems satisfy the integrability requirement, while systems linearisable through some more complicated methods need not. A most interesting result was the one obtained for the limit n → ∞ of the Gambier mapping. As explained, the interpretation, in the discrete case, was that the mapping does confine, albeit after an infinite number of steps 14 (which in principle can be interpreted as "non confinement") and in the light of this, the confinement property in the ultradiscrete case should not be astonishing. It is worth pointing out that we were able to construct the exact solution of the linearisable equations and thus make explicit the structure of their singularities (the analogous approach in the discrete case is much harder and in some cases next to impossible).
Just as with the singularity confinement test for discrete equations, the ultradiscrete singularity confinement furnishes only a local information about the solution which may be insufficient for the detection of integrability of the equation. For the latter it is important to combine the local singularity structure with the global behaviour of the solution. Such an approach has been already proposed by Halburd and Southall 16 in a "standard" ultradiscrete setting. It would be interesting to investigate the possibility of extending their approach to the case of ultradiscrete systems with parity variables.