Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾
An ultradiscrete system corresponding to the q-Painlevé equation of type A₆⁽¹⁾, which is a q-difference analogue of the second Painlevé equation, is proposed. Exact solutions with two parameters are constructed for the ultradiscrete system.
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irk-123456789-1471812019-02-14T01:26:34Z Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ Murata, M. An ultradiscrete system corresponding to the q-Painlevé equation of type A₆⁽¹⁾, which is a q-difference analogue of the second Painlevé equation, is proposed. Exact solutions with two parameters are constructed for the ultradiscrete system. 2011 Article Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ / M. Murata // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E17; 39A12 DOI:10.3842/SIGMA.2011.059 http://dspace.nbuv.gov.ua/handle/123456789/147181 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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An ultradiscrete system corresponding to the q-Painlevé equation of type A₆⁽¹⁾, which is a q-difference analogue of the second Painlevé equation, is proposed. Exact solutions with two parameters are constructed for the ultradiscrete system. |
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Murata, M. Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Murata, M. |
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Murata, M. |
| title |
Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ |
| title_short |
Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ |
| title_full |
Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ |
| title_fullStr |
Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ |
| title_full_unstemmed |
Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ |
| title_sort |
exact solutions with two parameters for an ultradiscrete painlevé equation of type a₆⁽¹⁾ |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/147181 |
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Exact Solutions with Two Parameters for an Ultradiscrete Painlevé Equation of Type A₆⁽¹⁾ / M. Murata // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 31 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT muratam exactsolutionswithtwoparametersforanultradiscretepainleveequationoftypea61 |
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2025-07-11T01:33:21Z |
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2025-07-11T01:33:21Z |
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1837312360525594624 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 059, 15 pages
Exact Solutions with Two Parameters
for an Ultradiscrete Painlevé Equation of Type A
(1)
6
?
Mikio MURATA
Department of Physics and Mathematics, College of Science and Engineering, Aoyama Gakuin
University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa, 252-5258 Japan
E-mail: murata@gem.aoyama.ac.jp
Received February 07, 2011, in final form June 11, 2011; Published online June 17, 2011
doi:10.3842/SIGMA.2011.059
Abstract. An ultradiscrete system corresponding to the q-Painlevé equation of type A
(1)
6 ,
which is a q-difference analogue of the second Painlevé equation, is proposed. Exact solutions
with two parameters are constructed for the ultradiscrete system.
Key words: Painlevé equations; ultradiscrete systems
2010 Mathematics Subject Classification: 33E17; 39A12
1 Introduction
Discrete Painlevé equations are prototype integrable systems studied from various points of
view [24, 28]. They are discrete equations which are reduced to the Painlevé equations in suitable
limiting processes, and moreover, which pass the singularity confinement test [4]. Many results
are already given about special solutions of discrete Painlevé equations [5, 11, 12, 13, 16, 25].
Ultradiscretization [30] is a limiting procedure transforming a given difference equation into
a cellular automaton. In addition the cellular automaton constructed by this procedure preserves
the essential properties of the original equation, such as the structure of exact solutions. In this
procedure, we first replace a dependent variable xn in a given equation by
xn = exp
(
Xn
ε
)
,
where ε is a positive parameter. Then, we apply ε log to both sides of the equation and take the
limit ε→ +0. Using identity
lim
ε→+0
ε log
(
eX/ε + eY/ε
)
= max (X,Y )
and exponential laws, we find that addition, multiplication, and division for the original variables
are replaced by maximum, addition, and subtraction for the new ones, respectively. In this way
the original difference equation is approximated to a piecewise linear equation which can be
regarded as a time evolution rule for a cellular automaton.
It is an interesting problem to study ultradiscrete analogues of the Painlevé equations and the
structure of their solutions. Some ultradiscrete Painlevé equations and their special solutions are
studied in, for example, [3, 8, 9, 10, 22, 26, 29]. However the structure of the general solutions
is completely unclear today.
?This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Spe-
cial Functions with Quantum Groups and Integrable Systems”. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSF.html
mailto:murata@gem.aoyama.ac.jp
http://dx.doi.org/10.3842/SIGMA.2011.059
http://www.emis.de/journals/SIGMA/OPSF.html
2 M. Murata
In this paper we propose a new ultradiscrete Painlevé equation of simultaneous type. With
this purpose, we start with a q-Painlevé equation of type A
(1)
6 (q-P (A6)) [5, 11, 12, 18, 19, 27, 28]
fnfn−1 = 1 + gn−1, gngn−1 =
aq2nfn
fn + qn
, (1.1)
where a and q are parameters. Equation (1.1) is the simplest nontrivial q-Painlevé equation
that admits a Bäcklund transformation. This equation is also referred to as q-analogue of the
second Painlevé equation
(fn+1fn − 1) (fnfn−1 − 1) =
aq2nfn
fn + qn
and reduced to the second Painlevé equation
d2y
ds2
= 2y3 + 2sy + c
in a continuous limit [23].
Furthermore, we propose an exact solution with two parameters for the ultradiscrete system.
Although the Painlevé equations and the q-analogues of these are not generally solvable in terms
of elementary functions [17, 18, 20, 31], it is an amazing fact that the ultradiscrete analogues of
these are “solvable”.
In Section 2, we present an ultradiscrete analogue of q-P (A6). In Section 3, we give an
exact solution with two parameters of this ultradiscrete system. In Section 4, we construct
an ultradiscrete Bäcklund transformation. The exact solutions with two parameters are also
obtained from a “seed” solution. In Section 5, we give ultradiscrete hypergeometric solutions
which are included in the solutions with two parameters. Finally concluding remarks are given
in Section 6.
2 Ultradiscrete Painlevé equation
We construct an ultradiscrete analogue of q-P (A6) (1.1). Let us introduce
fn = exp (Fn/ε) , gn = exp (Gn/ε) , q = exp (Q/ε) , a = exp (A/ε)
and take the limit ε → +0. Then q-P (A6) (1.1) is reduced to an ultradiscrete analogue of
q-P (A6) (ud-P (A6)),
Fn + Fn−1 = max (0, Gn−1) , (2.1a)
Gn +Gn−1 = A+ 2nQ−max (0, nQ− Fn) . (2.1b)
Because one cannot make a known second order single equation from this system, this ud-P (A6)
is an essentially new ultradiscrete Painlevé system.
In [6], we have given another ud-P (A6) by means of ultradiscretization with parity variables,
which is an extended version of ultradiscrete procedure. This procedure keeps track of the sign
of original variables [15]. We have also presented its special solution that corresponds to the
hypergeometric solution in the discrete system.
3 Solutions
In order to construct a solution of ud-P (A6), we take the following strategy. First we seek
solutions for linear systems which are obtained from the piecewise linear system. These solutions
satisfy ud-P (A6) in some restricted range of n. Next we connect these solutions together to
ensure that they satisfy (2.1) for any n.
Exact Solutions for an Ultradiscrete Painlevé Equation 3
Theorem 1. ud-P (A6) admits the following solution for Q > 0, A = 2(m + r)Q, m ∈ N,
−1/2 < r ≤ 1/2:
Fn = d1 (−1)n−m , Gn =
2n+ 2m+ 2r + 1
2
Q+ d2 (−1)n−m ,
for n ≤ −m− 1, where d1 and d2 satisfy
− (m+ 2)Q ≤ d1 ≤ (m+ 1)Q,
2r − 5
2
Q ≤ d2 ≤
3− 2r
2
Q;
Fn =
n+m+ r
2
Q+ e1 (−1)n−m − e2 (n−m) (−1)n−m ,
Gn =
2n+ 2m+ 2r + 1
2
Q+ e2 (−1)n−m ,
for −m ≤ n ≤ m− 1, where e1 and e2 satisfy
−1 + 2r
2
Q ≤ e2 ≤
3 + 2r
2
Q, e1 + e2 ≤
1 + r
2
Q, e1 + 2e2 ≥ −
2 + r
2
Q,
e1 + (2m− 1) e2 ≤
2m+ r − 1
2
Q, e1 + 2me2 ≥ −
2m+ r
2
Q,
and
Fn =
n+ 2m+ 2r
3
Q+ h1 cos
2
3
π (n−m) +
2h2 − h1√
3
sin
2
3
π (n−m) ,
Gn =
2n+ 4m+ 4r + 1
3
Q+ h2 cos
2
3
π (n−m) +
h2 − 2h1√
3
sin
2
3
π (n−m) ,
for n ≥ m, where h1 and h2 satisfy
h1 ≤
6− 2r
3
Q, h2 ≥
2r − 4
3
Q, h2 − h1 ≤
2− 2r
3
Q.
Here the relations between d1, d2 and e1, e2 are
d1 =
r
2
Q+ e1 + 2me2 − 2 max
(
0,
2r − 1
2
Q− e2
)
, d2 = e2,
and those between e1, e2 and h1, h2 are
h1 = −r
6
Q+ e1, h2 =
1− 2r
6
Q+ e2 −max
(
0,−r
2
Q− e1
)
.
Proof. We consider the case A = 2(m + r)Q, m ∈ N and −1/2 < r ≤ 1/2. If Gn−1 ≤ 0 and
nQ− Fn ≤ 0, then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = 0, Gn +Gn−1 = (2n+ 2m+ 2r)Q. (3.1)
The general solution to the linear system (3.1) is
Fn = d1 (−1)n−m , Gn =
2n+ 2m+ 2r + 1
2
Q+ d2 (−1)n−m , (3.2)
where d1 and d2 are arbitrary constants. If d1 = d2 = 0, the particular solution (3.2) satisfies
Gn−1 ≤ 0 and nQ − Fn ≤ 0 for n ≤ −m − 1. The sufficient condition that the general
solution (3.2) satisfies Gn−1 ≤ 0 and nQ− Fn ≤ 0 for n ≤ −m− 1 is
− (m+ 2)Q ≤ d1 ≤ (m+ 1)Q,
2r − 5
2
Q ≤ d2 ≤
3− 2r
2
Q. (3.3)
4 M. Murata
Therefore (3.2) that satisfies (3.3) is a solution to ud-P (A6) for n ≤ −m− 1. If Gn−1 ≥ 0 and
nQ− Fn ≤ 0, then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = Gn−1, Gn +Gn−1 = (2n+ 2m+ 2r)Q. (3.4)
The general solution to the linear system (3.4) is
Fn =
n+m+ r
2
Q+ e1 (−1)n−m − e2 (n−m) (−1)n−m ,
Gn =
2n+ 2m+ 2r + 1
2
Q+ e2 (−1)n−m , (3.5)
where e1 and e2 are arbitrary constants. If e1 = e2 = 0, (3.5) satisfies Gn ≥ 0 and nQ−Fn ≤ 0
for −m ≤ n ≤ m − 1. The condition that the general solution (3.5) satisfies Gn ≥ 0 and
nQ− Fn ≤ 0 for −m ≤ n ≤ m− 1 is
−1 + 2r
2
Q ≤ e2 ≤
3 + 2r
2
Q, e1 + e2 ≤
1 + r
2
Q, e1 + 2e2 ≥ −
2 + r
2
Q,
e1 + (2m− 1) e2 ≤
2m+ r − 1
2
Q, e1 + 2me2 ≥ −
2m+ r
2
Q. (3.6)
Therefore (3.5) that satisfies (3.6) is a solution to ud-P (A6) for −m ≤ n ≤ m− 1. If Gn−1 ≥ 0
and nQ−Fn ≥ 0, then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = Gn−1, Gn +Gn−1 = (n+ 2m+ 2r)Q+ Fn. (3.7)
The general solution to the linear system (3.7) is
Fn =
n+ 2m+ 2r
3
Q+ h1 cos
2
3
π (n−m) +
2h2 − h1√
3
sin
2
3
π (n−m) ,
Gn =
2n+ 4m+ 4r + 1
3
Q+ h2 cos
2
3
π (n−m) +
h2 − 2h1√
3
sin
2
3
π (n−m) , (3.8)
where h1 and h2 are arbitrary constants. If h1 = h2 = 0, (3.8) satisfiesGn−1 ≥ 0 and nQ−Fn ≥ 0
for n ≥ m+1. The condition that the general solution (3.8) satisfies Gn−1 ≥ 0 and nQ−Fn ≥ 0
for n ≥ m+ 1 is
h1 ≤
6− 2r
3
Q, h2 ≥
2r − 4
3
Q, h2 − h1 ≤
2− 2r
3
Q. (3.9)
Therefore (3.8) that satisfies (3.9) is a solution to ud-P (A6) for n ≥ m + 1. The relations
between d1, d2 and e1, e2 can be obtained from (2.1a) for n = −m:
F−m + F−m−1 = max (0, G−m−1) ,
(3.2) for n = −m− 1:
F−m−1 = −d1, G−m−1 =
2r − 1
2
Q− d2,
and (3.5) for n = −m,−m− 1 respectively:
F−m =
r
2
Q+ 2me2 + e1, G−m−1 =
2r − 1
2
Q− e2.
We have
d1 =
r
2
Q+ e1 + 2me2 − 2 max
(
0,
2r − 1
2
Q− e2
)
, d2 = e2.
Exact Solutions for an Ultradiscrete Painlevé Equation 5
Moreover the relations between e1, e2 and h1, h2 can be obtained from (2.1b) for n = m:
Gm +Gm−1 = (4m+ 2r)Q−max (0,mQ− Fm) ,
(3.5) for n = m,m− 1 respectively:
Fm =
2m+ r
2
Q+ e1, Gm−1 =
4m+ 2r − 1
2
Q− e2,
and (3.8) for n = m:
Fm =
3m+ 2r
3
Q+ h1, Gm =
6m+ 4r + 1
3
Q+ h2.
And we have
h1 = −r
6
Q+ e1, h2 =
1− 2r
6
Q+ e2 −max
(
0,−r
2
Q− e1
)
.
When |e1| and |e2| are sufficiently small, we shall write “e1 ∼ 0, e2 ∼ 0” as an abbreviation, If
e1 ∼ 0 and e2 ∼ 0, then we find that
d1 ∼
r
2
Q, d2 ∼ 0
satisfy (3.3), and
h1 ∼ −
r
6
Q, h2 ∼
1− 2r
6
Q−max
(
0,−r
2
Q
)
satisfy (3.9). Therefore we have Theorem 1 by connecting these solutions together. �
Theorem 2. ud-P (A6) admits the following solution for Q > 0, A = 2(m + r)Q, −m ∈ N,
0 < r ≤ 1/2:
Fn = d1 (−1)n , Gn =
2n+ 2m+ 2r + 1
2
Q+ d2 (−1)n
for n ≤ −1, where d1 and d2 satisfy
−2Q ≤ d1 ≤ Q,
2m+ 2r − 1
2
Q ≤ d2 ≤
−2m− 2r + 3
2
Q;
Fn = e1 (−1)n , Gn =
2n+ 4m+ 4r + 1
4
Q+ e1n (−1)n + e2 (−1)n
for 0 ≤ n ≤ −2m− 1, where e1 and e2 satisfy
−Q ≤ e1 ≤ 2Q, e2 ≤ −
4m+ 4r + 1
4
Q, e1 + e2 ≥
4m+ 4r + 3
4
Q,
− (2m+ 2) e1 + e2 ≤
3− 4r
4
Q, − (2m+ 3) e1 + e2 ≥
4r − 5
4
Q,
and
Fn =
n+ 2m+ 2r
3
Q+ h1 cos
2
3
π (n+ 2m) +
2h2 − h1√
3
sin
2
3
π (n+ 2m) ,
Gn =
2n+ 4m+ 4r + 1
3
Q+ h2 cos
2
3
π (n+ 2m) +
h2 − 2h1√
3
sin
2
3
π (n+ 2m)
6 M. Murata
for n ≥ −2m, where h1 and h2 satisfy
h1 ≤
4r + 3
3
Q, h2 ≥ −
4r + 1
3
Q, h2 − h1 ≤
4r + 5
3
Q.
Here the relations between d1, d2 and e1, e2 are
d1 = e1, d2 = −1
4
Q+ e2 + max (0,−e1) ,
and those between e1, e2 and h1, h2 are
h1 = −2r
3
Q+ e1 + max
{
0,
4r − 1
4
Q+ (2m+ 1) e1 − e2
}
,
h2 = −4r + 1
12
Q− 2me1 + e2 + max
{
0,
4r − 1
4
Q+ (2m+ 1) e1 − e2
}
.
Theorem 3. ud-P (A6) admits the following solution for Q > 0, A = 2(m + r)Q, −m ∈ N,
−1/2 < r ≤ 0:
Fn = d1 (−1)n , Gn =
2n+ 2m+ 2r + 1
2
Q+ d2 (−1)n
for n ≤ −1, where d1 and d2 satisfy
−2Q ≤ d1 ≤ Q,
2m+ 2r − 1
2
Q ≤ d2 ≤
−2m− 2r + 3
2
Q;
Fn = e1 (−1)n , Gn =
2n+ 4m+ 4r + 1
4
Q+ e1n (−1)n + e2 (−1)n
for 0 ≤ n ≤ −2m, where e1 and e2 satisfy
−Q ≤ e1 ≤ 2Q, e2 ≤ −
4m+ 4r + 1
4
Q, e1 + e2 ≥
4m+ 4r + 3
4
Q,
− (2m+ 1) e1 + e2 ≥
4r − 1
4
Q, − (2m+ 2) e1 + e2 ≤
3− 4r
4
Q,
and
Fn =
n+ 2m+ 2r
3
Q+ h1 cos
2
3
π (n+ 2m) +
2h2 − h1√
3
sin
2
3
π (n+ 2m) ,
Gn =
2n+ 4m+ 4r + 1
3
Q+ h2 cos
2
3
π (n+ 2m) +
h2 − 2h1√
3
sin
2
3
π (n+ 2m)
for n ≥ −2m+ 1, where h1 and h2 satisfy
h1 ≤
4r + 3
3
Q, h2 ≥ −
4r + 7
3
Q, h2 − h1 ≤
4r + 5
3
Q.
Here the relations between d1, d2 and e1, e2 are
d1 = e1, d2 = −1
4
Q+ e2 + max (0,−e1) ,
and those between e1, e2 and h1, h2 are
h1 =
4r + 3
12
Q− (2m− 1) e1 + e2 −max
(
0,
4r + 1
4
Q− 2me1 + e2
)
,
h2 = −4r + 1
12
Q− 2me1 + e2.
Exact Solutions for an Ultradiscrete Painlevé Equation 7
Proof. We consider the case A = 2(m+ r)Q, −m ∈ N and −1/2 < r ≤ 1/2. If Gn−1 ≤ 0 and
nQ− Fn ≤ 0, then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = 0, Gn +Gn−1 = (2n+ 2m+ 2r)Q. (3.10)
The general solution to the linear system (3.10) is
Fn = d1 (−1)n , Gn =
2n+ 2m+ 2r + 1
2
Q+ d2 (−1)n , (3.11)
where d1 and d2 are arbitrary constants. If d1 = d2 = 0, the particular solution (3.11) satisfies
Gn ≤ 0 and nQ − Fn ≤ 0 for n ≤ −1. The condition that the general solution (3.11) satisfies
Gn−1 ≤ 0 and nQ− Fn ≤ 0 for n ≤ −1 is
−2Q ≤ d1 ≤ Q,
2m+ 2r − 1
2
Q ≤ d2 ≤
−2m− 2r + 3
2
Q. (3.12)
Therefore (3.11) that satisfies (3.12) is a solution to ud-P (A6) for n ≤ −1. If Gn−1 ≤ 0 and
nQ− Fn ≥ 0, then (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = 0, Gn +Gn−1 = (n+ 2m+ 2r)Q+ Fn. (3.13)
The general solution to the linear system (3.13) is
Fn = e1 (−1)n , Gn =
2n+ 4m+ 4r + 1
4
Q+ e1n (−1)n + e2 (−1)n , (3.14)
where e1 and e2 are arbitrary constants. If e1 = e2 = 0 and 0 < r ≤ 1/2, (3.14) satisfies
Gn−1 ≤ 0 and nQ−Fn ≥ 0 for 1 ≤ n ≤ −2m−1. The condition that the general solution (3.14)
satisfies Gn−1 ≤ 0 and nQ− Fn ≥ 0 for 1 ≤ n ≤ −2m− 1 is
−Q ≤ e1 ≤ 2Q, e2 ≤ −
4m+ 4r + 1
4
Q, e1 + e2 ≥
4m+ 4r + 3
4
Q,
− (2m+ 2) e1 + e2 ≤
3− 4r
4
Q, − (2m+ 3) e1 + e2 ≥
4r − 5
4
Q. (3.15)
Therefore (3.14) that satisfies (3.15) is a solution to ud-P (A6) for 1 ≤ n ≤ −2m−1. If e1 = e2 =
0 and −1/2 < r ≤ 0, then (3.14) satisfies Gn−1 ≤ 0 and nQ − Fn ≥ 0 for 1 ≤ n ≤ −2m. The
condition that the general solution (3.14) satisfies Gn−1 ≤ 0 and nQ−Fn ≥ 0 for 1 ≤ n ≤ −2m
is
−Q ≤ e1 ≤ 2Q, e2 ≤ −
4m+ 4r + 1
4
Q, e1 + e2 ≥
4m+ 4r + 3
4
Q,
− (2m+ 1) e1 + e2 ≥
4r − 1
4
Q, − (2m+ 2) e1 + e2 ≤
3− 4r
4
Q. (3.16)
Therefore (3.14) that satisfies (3.16) is a solution to ud-P (A6) for 1 ≤ n ≤ −2m. If Gn−1 ≥ 0
and nQ−Fn ≥ 0, then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = Gn−1, Gn +Gn−1 = (n+ 2m+ 2r)Q+ Fn. (3.17)
The general solution to the linear system (3.17) is
Fn =
n+ 2m+ 2r
3
Q+ h1 cos
2
3
π (n+ 2m) +
2h2 − h1√
3
sin
2
3
π (n+ 2m) ,
Gn =
2n+ 4m+ 4r + 1
3
Q+ h2 cos
2
3
π (n+ 2m) +
h2 − 2h1√
3
sin
2
3
π (n+ 2m) , (3.18)
8 M. Murata
where h1 and h2 are arbitrary constants. If h1 = h2 = 0 and 0 < r ≤ 1/2, (3.18) satisfies the
conditions Gn ≥ 0 and nQ−Fn ≥ 0 for n ≥ −2m. The condition that the general solution (3.18)
satisfies Gn ≥ 0 and nQ− Fn ≥ 0 for n ≥ −2m is
h1 ≤
4r + 3
3
Q, h2 ≥ −
4r + 1
3
Q, h2 − h1 ≤
4r + 5
3
Q. (3.19)
Therefore (3.18) that satisfies (3.19) is a solution to ud-P (A6) for n ≥ −2m. If h1 = h2 = 0
and −1/2 < r ≤ 0, (3.18) satisfies Gn ≥ 0 and nQ − Fn ≥ 0 for n ≥ −2m + 1. The condition
that the general solution (3.18) satisfies Gn ≥ 0 and nQ− Fn ≥ 0 for n ≥ −2m+ 1 is
h1 ≤
4r + 3
3
Q, h2 ≥ −
4r + 7
3
Q, h2 − h1 ≤
4r + 5
3
Q. (3.20)
Therefore (3.18) that satisfies (3.20) is a solution to ud-P (A6) for n ≥ −2m+ 1. The relations
between d1, d2 and e1, e2 can be obtained from (2.1b) for n = 0:
G0 +G−1 = (2m+ 2r)Q−max (0,−F0) ,
(3.11) for n = 0,−1 respectively:
F0 = d1, G−1 =
2m+ 2r − 1
2
Q− d2,
and (3.14) for n = 0:
F0 = e1, G0 =
4m+ 4r + 1
4
Q+ e2.
We have
d1 = e1, d2 = −1
4
Q+ e2 + max (0,−e1) .
Moreover in the case 0 < r ≤ 1/2, the relations between e1, e2 and h1, h2 can be obtained
from (2.1a) for n = −2m:
F−2m + F−2m−1 = max (0, G−2m−1) ,
(3.14) for n = −2m− 1:
F−2m−1 = −e1, G−2m−1 =
4r − 1
4
Q+ (2m+ 1) e1 − e2,
and (3.18) for n = −2m,−2m− 1 respectively:
F−2m =
2r
3
Q+ h1, G−2m−1 =
4r − 1
3
Q+ h1 − h2.
We have
h1 = −2r
3
Q+ e1 + max
{
0,
4r − 1
4
Q+ (2m+ 1) e1 − e2
}
,
h2 = −4r + 1
12
Q− 2me1 + e2 + max
{
0,
4r − 1
4
Q+ (2m+ 1) e1 − e2
}
.
In the case −1/2 < r ≤ 0, the relations between e1, e2 and h1, h2 can be obtained from (2.1a)
for n = −2m+ 1:
F−2m+1 + F−2m = max (0, G−2m) ,
Exact Solutions for an Ultradiscrete Painlevé Equation 9
(3.14) for n = −2m:
F−2m = e1, G−2m =
4r + 1
4
Q− 2me1 + e2,
and (3.18) for n = −2m+ 1,−2m respectively:
F−2m+1 =
2r + 1
3
Q− h1 + h2, G−2m =
4r + 1
3
Q+ h2.
We have
h1 =
4r + 3
12
Q− (2m− 1) e1 + e2 −max
(
0,
4r + 1
4
Q− 2me1 + e2
)
,
h2 = −4r + 1
12
Q− 2me1 + e2.
If e1 ∼ 0, e2 ∼ 0, then we find that
d1 ∼ 0, d2 ∼ −
1
4
Q
satisfy (3.12),
h1 ∼ −
2r
3
Q+ max
(
0,
4r − 1
4
Q
)
, h2 ∼ −
4r + 1
12
Q+ max
(
0,
4r − 1
4
Q
)
satisfy (3.19), and
h1 ∼
4r + 3
12
Q−max
(
0,
4r + 1
4
Q
)
, h2 ∼ −
4r + 1
12
Q
satisfy (3.20). We have Theorem 2 and Theorem 3 by connecting these solutions together. �
Theorem 4. ud-P (A6) admits the following solution for Q > 0, A = 2rQ, −1/2 < r ≤ 1/2:
Fn = d1 (−1)n , Gn =
2n+ 2r + 1
2
Q+ d2 (−1)n ,
for n ≤ −1, where d1 and d2 satisfy
−2Q ≤ d1 ≤ Q,
2r − 5
2
Q ≤ d2 ≤
3− 2r
2
Q,
and
Fn =
n+ 2r
3
Q+ h1 cos
2
3
πn+
2h2 − h1√
3
sin
2
3
πn,
Gn =
2n+ 4r + 1
3
Q+ h2 cos
2
3
πn+
h2 − 2h1√
3
sin
2
3
πn,
for n ≥ 1, where h1 and h2 satisfy
h1 ≤
4r + 3
3
Q, h2 ≥
2r − 4
3
Q, h2 − h1 ≤
2− 2r
3
Q.
Here the relations between d1, d2 and F0, G0 are
d1 = F0 −max {0, 2rQ−G0 −max (0,−F0)} , d2 = −2r + 1
2
Q+G0 + max (0,−F0) ,
and those between h1, h2 and F0, G0 are
h1 = −2r
3
Q+ F0 −max (0,−G0) , h2 = G0 −
4r + 1
3
Q.
10 M. Murata
Proof. We consider the case A = 2rQ and −1/2 < r ≤ 1/2. If Gn−1 ≤ 0 and nQ − Fn ≤ 0,
then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = 0, Gn +Gn−1 = (2n+ 2r)Q. (3.21)
The general solution to the linear system (3.21) is
Fn = d1 (−1)n , Gn =
2n+ 2r + 1
2
Q+ d2 (−1)n , (3.22)
where d1 and d2 are arbitrary constants. If d1 = d2 = 0, the particular solution (3.22) satisfies
Gn−1 ≤ 0 and nQ−Fn ≤ 0 for n ≤ −1. The sufficient condition that the general solution (3.22)
satisfies Gn−1 ≤ 0 and nQ− Fn ≤ 0 for n ≤ −1 is
−2Q ≤ d1 ≤ Q,
2r − 5
2
Q ≤ d2 ≤
3− 2r
2
Q. (3.23)
Therefore (3.22) that satisfies (3.23) is a solution to ud-P (A6) for n ≤ −1. If Gn−1 ≥ 0 and
nQ− Fn ≥ 0, then ud-P (A6) (2.1) can be written as the following system of linear equations:
Fn + Fn−1 = Gn−1, Gn +Gn−1 = (n+ 2r)Q+ Fn. (3.24)
The general solution to the linear system (3.24) is
Fn =
n+ 2r
3
Q+ h1 cos
2
3
πn+
2h2 − h1√
3
sin
2
3
πn,
Gn =
2n+ 4r + 1
3
Q+ h2 cos
2
3
πn+
h2 − 2h1√
3
sin
2
3
πn, (3.25)
where h1 and h2 are arbitrary constants. If h1 = h2 = 0, (3.25) satisfies Gn ≥ 0 and nQ−Fn ≥ 0
for n ≥ 1. The condition that the general solution (3.25) satisfies Gn ≥ 0 and nQ− Fn ≥ 0 for
n ≥ 1 is
h1 ≤
4r + 3
3
Q, h2 ≥
2r − 4
3
Q, h2 − h1 ≤
2− 2r
3
Q. (3.26)
Therefore (3.25) that satisfies (3.26) is a solution to ud-P (A6) for n ≥ 2. The relations between
d1, d2 and F0, G0 can be obtained from (2.1) for n = 0:
F0 + F−1 = max (0, G−1) , G0 +G−1 = 2rQ−max (0,−F0) ,
and (3.22) for n = −1:
F−1 = −d1, G−1 =
2r − 1
2
Q− d2.
We have
d1 = F0 −max {0, 2rQ−G0 −max (0,−F0)} , d2 = −2r + 1
2
Q+G0 + max (0,−F0) .
Moreover the relations between h1, h2 and F0, G0 can be obtained from (2.1a) for n = 1:
F1 + F0 = max (0, G0) ,
and (3.25) for n = 1, 0 respectively:
F1 =
2r + 1
3
Q− h1 + h2, G0 =
4r + 1
3
Q+ h2.
Exact Solutions for an Ultradiscrete Painlevé Equation 11
And we have
h1 = −2r
3
Q+ F0 −max (0,−G0) , h2 = G0 −
4r + 1
3
Q.
If F0 ∼ 0 and G0 ∼ 0, then we find that
d1 ∼ −max (0, 2rQ) , d2 ∼ −
2r + 1
2
Q
satisfy (3.23), and
h1 ∼ −
2r
3
Q, h2 ∼ −
4r + 1
3
Q
satisfy (3.26). Therefore we have Theorem 4 by connecting these solutions together. �
The exact solutions with two parameters for any parameter A have been given in this section.
4 Bäcklund transformation
q-P (A6) have the Bäcklund transformation [5, 28]. That is, if fn and gn satisfy q-P (A6) (1.1),
then
fn =
qn
gn
aqn+1fn+1 + gn
qnfn+1 + gn
, gn =
qn+1
fn+1
aqn+1fn+1 + gn
qnfn+1 + gn
(4.1)
satisfy q-P (A6):
fnfn−1 = 1 + gn−1, gngn−1 =
aq2q2nfn
fn + qn
,
and
fn+1 =
qn+1
gn
aqnfn + gn
qn+1fn + gn
, gn =
qn
fn
aqnfn + gn
qn+1fn + gn
(4.2)
also satisfy q-P (A6):
fnfn−1 = 1 + gn−1, gngn−1 =
aq−2q2nfn
fn + qn
.
So we apply the procedure of the ultradiscretization to (4.1) and (4.2). Then we have the
following theorems.
Theorem 5. If Fn and Gn satisfy ud-P (A6) (2.1), then
Fn = max {Fn+1 + (n+ 1)Q+A−Gn, 0} −max (Fn+1, Gn − nQ) ,
Gn = Q+ max {(n+ 1)Q+A,Gn − Fn+1} −max (Fn+1, Gn − nQ)
satisfy ud-P (A6):
Fn + Fn−1 = max (0,Gn−1) , Gn + Gn−1 = A+ 2Q+ 2nQ−max (0, nQ− Fn) .
12 M. Murata
Proof. We can obtain
Fn = max {Fn+1 + (n+ 1)Q+A−Gn, 0} −max (Fn+1, Gn − nQ)
= nQ−Gn + max {A+ (n+ 1)Q+ max (0, Gn) , Fn +Gn}
−max {nQ+ max (0, Gn) , Fn +Gn} ,
Gn = Q+ max {(n+ 1)Q+A,Gn − Fn+1} −max (Fn+1, Gn − nQ)
= (n+ 1)Q+ Fn −max (0, Gn) + max {A+ (n+ 1)Q+ max (0, Gn) , Fn +Gn}
−max {nQ+ max (0, Gn) , Fn +Gn}
by using (2.1a), and
Fn−1 = max (Fn + nQ+A−Gn−1, 0)−max {Fn, Gn−1 − (n− 1)Q}
= Gn − nQ+ max (Fn, nQ)− Fn + max {Gn + max (Fn, nQ) , nQ}
−max {Gn + max (Fn, nQ) , A+ (n+ 1)Q} ,
Gn−1 = Q+ max {nQ+A,Gn−1 − Fn} −max {Fn, Gn−1 − (n− 1)Q}
= A+ (n+ 1)Q− Fn + max {Gn + max (Fn, nQ) , nQ}
−max {Gn + max (Fn, nQ) , A+ (n+ 1)Q}
by using (2.1b). Thus we find
Fn + Fn−1 = max (0,Gn−1)
= max (Fn, nQ)− Fn + max {A+ (n+ 1)Q+ max (0, Gn) , Fn +Gn}
−max {Gn + max (Fn, nQ) , A+ (n+ 1)Q} ,
Gn + Gn−1 = A+ 2Q+ 2nQ−max (0, nQ− Fn) = A+ (2n+ 2)Q−max (0, Gn)
+ max {A+ (n+ 1)Q+ max (0, Gn) , Fn +Gn}
−max {Gn + max (Fn, nQ) , A+ (n+ 1)Q} . �
Theorem 6. If Fn and Gn satisfy ud-P (A6) (2.1), then
Fn+1 = max (nQ+A+ Fn −Gn, 0)−max {Fn, Gn − (n+ 1)Q} ,
Gn = −Q+ max (nQ+A,Gn − Fn)−max {Fn, Gn − (n+ 1)Q}
satisfy ud-P (A6):
Fn + Fn−1 = max (0, Gn−1) , Gn + Gn−1 = A− 2Q+ 2nQ−max (0, nQ− Fn) .
Proof. We can obtain
Fn−1 = max {(n− 2)Q+A+ Fn−2 −Gn−2, 0} −max {Fn−2, Gn−2 − (n− 1)Q} ,
= (n− 1)Q−Gn−2 + max {A+ (n− 2)Q+ max (0, Gn−2) , Fn−1 +Gn−2}
−max {(n− 1)Q+ max (0, Gn−2) , Fn−1 +Gn−2}
by using (2.1a), and
Fn = max {(n− 1)Q+A+ Fn−1 −Gn−1, 0} −max (Fn−1, Gn−1 − nQ)
= Gn−2 − (n− 1)Q+ max {Fn−1, (n− 1)Q} − Fn−1
+ max [Gn−2 + max {Fn−1, (n− 1)Q} , (n− 1)Q]
−max [Gn−2 + max {Fn−1, (n− 1)Q} , A+ (n− 2)Q] ,
Gn−1 = −Q+ max {(n− 1)Q+A,Gn−1 − Fn−1} −max (Fn−1, Gn−1 − nQ)
Exact Solutions for an Ultradiscrete Painlevé Equation 13
= A+ (n− 2)Q− Fn−1 + max [Gn−2 + max {Fn−1, (n− 1)Q} , (n− 1)Q]
−max [Gn−2 + max {Fn−1, (n− 1)Q} , A+ (n− 2)Q]
by using (2.1b). Thus we find
Fn + Fn−1 = max (0, Gn−1) = max {Fn−1, (n− 1)Q} − Fn−1
+ max {A+ (n− 2)Q+ max (0, Gn−2) , Fn−1 +Gn−2}
−max [Gn−2 + max {Fn−1, (n− 1)Q} , A+ (n− 2)Q] .
We obtain
Fn = max {(n− 1)Q+A+ Fn−1 −Gn−1, 0} −max (Fn−1, Gn−1 − nQ)
= nQ−Gn−1 + max {A+ (n− 1)Q+ max (0, Gn−1) , Fn +Gn−1}
−max {nQ+ max (0, Gn−1) , Fn +Gn−1} ,
Gn−1 = −Q+ max {(n− 1)Q+A,Gn−1 − Fn−1} −max (Fn−1, Gn−1 − nQ)
= (n− 1)Q+ Fn −max (0, Gn−1)
+ max {A+ (n− 1)Q+ max (0, Gn−1) , Fn +Gn−1}
−max {nQ+ max (0, Gn−1) , Fn +Gn−1}
by using (2.1a), and
Gn = −Q+ max (nQ+A,Gn − Fn)−max {Fn, Gn − (n+ 1)Q}
= A+ (n− 1)Q− Fn + max {Gn−1 + max (Fn, nQ) , nQ}
−max {Gn−1 + max (Fn, nQ) , A+ (n− 1)Q}
by using (2.1b). Thus we find
Gn + Gn−1 = A− 2Q+ 2nQ−max (0, nQ− Fn) = A+ (2n− 2)Q−max (0, Gn−1)
+ max {A+ (n− 1)Q+ max (0, Gn−1) , Fn +Gn−1}
−max {Gn−1 + max (Fn, nQ) , A+ (n− 1)Q} . �
So the exact solutions also can be obtained from the solution in Theorem 4 by using the Bäcklund
transformation.
5 Special solutions
In [5], Hamamoto, Kajiwara and Witte constructed hypergeometric solutions to q-P (A6) by
applying Bäcklund transformations to the “seed” solution which satisfies a Riccati equation.
Their solutions have a determinantal form with basic hypergeometric function elements whose
continuous limits are showed by them to be Airy functions, the hypergeometric solutions of
the second Painlevé equation. In [18, 19], S. Nishioka proved that transcendental solutions
of q-P (A6) in a decomposable extension may exist only for special parameters, and that each of
them satisfies the Riccati equation mentioned above if we apply the Bäcklund transformations
to it appropriate times. He also proved non-existence of algebraic solutions.
q-P (A6) (1.1) for a = q2m+1 (m ∈ Z) has the hypergeometric solution. The case of A =
(2m+ 1)Q in ud-P (A6) corresponds to a = q2m+1 in the discrete system. It is hard to apply
the ultradiscretization procedure to the hypergeometric series. However according to [22], an
ultradiscrete hypergeometric solution is given in terms of nQ and (−1)nQ. If h1 = h2 = 0 and
14 M. Murata
r = 1/2 in Theorem 4, then we obtain an ultradiscrete hypergeometric solution of ud-P (A6) for
A = Q:
Fn =
{
1
3Q (−1)n (n ≤ −1),
n+1
3 Q (n ≥ 0),
Gn =
{
(n+ 1)Q (n ≤ −1),
2n+3
3 Q (n ≥ 0).
If h1 = h2 = 0 and r = 1/2 in Theorem 1, then we obtain an ultradiscrete hypergeometric
solution of ud-P (A6) for A = (2m+ 1)Q (m ∈ N):
Fn =
1
3Q (−1)n+m (n ≤ −m− 1),
2n+2m+1
4 Q+ 1
12Q (−1)n−m (−m ≤ n ≤ m− 1),
n+2m+1
3 Q (n ≥ m),
Gn =
{
(n+m+ 1)Q (n ≤ m− 1),
2n+4m+3
3 Q (n ≥ m).
If h1 = h2 = 0 and r = 1/2 in Theorem 2, then we have an ultradiscrete hypergeometric solution
for A = (2m+ 1)Q (−m ∈ N):
Fn =
{
0 (n ≤ −2m− 1),
n+2m+1
3 Q (n ≥ −2m),
Gn =
(n+m+ 1)Q (n ≤ −1),
2n+4m+3
4 Q− 1
12Q (−1)n (0 ≤ n ≤ −2m− 1),
2n+4m+3
3 Q (n ≥ −2m).
6 Concluding remarks
We have given the ultradiscrete analogue of q-P (A6). Moreover, we have presented the exact
solutions with two parameters. These solutions are expressed by using linear functions and
periodic functions. But the exact solution is only useful when the two parameters are in a limited
range. If one wants to construct the exact solution for any initial values, then one needs to use
a multitude of branches with respect to n in order to express a solution. We have also presented
its special solutions that correspond to the hypergeometric solutions of q-P (A6). The ultra-
discrete hypergeometric solutions are included in the resulting solutions with two parameters.
There are many studies on analytic properties of solutions to the Painlevé equations [1, 2, 7].
But there exist few studies on analytic properties of the q-Painlevé equations [14, 21]. We hope
to study the q-Painlevé equations by employing the results in the ultradiscrete systems.
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http://dx.doi.org/10.1103/PhysRevLett.76.3247
1 Introduction
2 Ultradiscrete Painlevé equation
3 Solutions
4 Bäcklund transformation
5 Special solutions
6 Concluding remarks
References
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