A Class of Special Solutions for the Ultradiscrete Painleve II Equation

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 and their special solutions are discussed, for example, in [2,3,4]. However, the class of solutions for ultradiscrete 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),

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. These solutions are, from their construction, considered to be counterparts of those of q-PII written by the determinants. Finally, concluding remarks are given in Section 4.

We start with a q-difference analog of the Airy equation

which reduces to the Airy equation

In order to ultradiscretize (3), we put τ = q m and q = e Q/ε (Q < 0). Furthermore, we introduce an ansatz for p-ultradiscretization, w(q m ) = {s(ω m )s(-ω m )}e Wm/ε , where ω m ∈ {+1, -1} denotes the sign of w(q m ) and s(ω) is defined by

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

where F m := mQ + W m -W m-1 . 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

and the other is a Bi-function-type for W 0 = (+1, 0) and

They show similar behavior as those of the Ai and Bi functions, respectively.

For the following discussion, we first introduce the results for (2). It has been shown in [10] that

, N < 0

(5) 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

For ( 6) and ( 7), we have their ultradiscrete analogs

and

respectively. For ( 8) and ( 9), we have

,

,

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 solutio

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