Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations
Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon eq...
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Kondo, K. 2019-02-21T07:13:39Z 2019-02-21T07:13:39Z 2013 Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations / K. Kondo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 39A12 DOI: http://dx.doi.org/10.3842/SIGMA.2013.068 https://nasplib.isofts.kiev.ua/handle/123456789/149360 Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations. The author would like to express his gratitude to Professor Tetsuji Tokihiro, who provided precise advices with a fine prospect. The author is also grateful to Professor Ralph Willox, who provided helpful comments for refining the results. In addition, the author thanks the anonymous referees for carefully reading the paper and giving many suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations Article published earlier |
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Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
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Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations Kondo, K. |
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Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
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Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
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Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
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Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
| title_sort |
ultradiscrete sine-gordon equation over symmetrized max-plus algebra, and noncommutative discrete and ultradiscrete sine-gordon equations |
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Kondo, K. |
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Kondo, K. |
| publishDate |
2013 |
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English |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
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Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.
|
| issn |
1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/149360 |
| citation_txt |
Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations / K. Kondo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. |
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AT kondok ultradiscretesinegordonequationoversymmetrizedmaxplusalgebraandnoncommutativediscreteandultradiscretesinegordonequations |
| first_indexed |
2025-11-26T06:40:17Z |
| last_indexed |
2025-11-26T06:40:17Z |
| _version_ |
1850615711079071744 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 068, 39 pages
Ultradiscrete sine-Gordon Equation
over Symmetrized Max-Plus Algebra,
and Noncommutative Discrete
and Ultradiscrete sine-Gordon Equations
Kenichi KONDO
5-13-12-207 Matsubara, Setagaya-ku, Tokyo 156-0043, Japan
E-mail: incidence algebra@poset.jp
Received January 08, 2013, in final form October 31, 2013; Published online November 12, 2013
http://dx.doi.org/10.3842/SIGMA.2013.068
Abstract. Ultradiscretization with negative values is a long-standing problem and several
attempts have been made to solve it. Among others, we focus on the symmetrized max-plus
algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultra-
discretization of the discrete sine-Gordon equation has already been proposed by previous
studies, but the equation and the solutions obtained here are considered to directly corre-
spond to the discrete counterpart. We also propose a noncommutative discrete analogue
of the sine-Gordon equation, reveal its relations to other integrable systems including the
noncommutative discrete KP equation, and construct multisoliton solutions by a repeated
application of Darboux transformations. Moreover, we derive a noncommutative ultradis-
crete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using
the symmetrized max-plus algebra. As a result, we have a complete set of commutative and
noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.
Key words: ultradiscrete sine-Gordon equation; symmetrized max-plus algebra; noncommu-
tative discrete sine-Gordon equation; noncommutative ultradiscrete sine-Gordon equation
2010 Mathematics Subject Classification: 37K10; 39A12
1 Introduction
1.1 Ultradiscretization and its problem
Ultradiscrete integrable systems are integrable systems where independent variables take values
in Z, and dependent variables in the max-plus algebra Rmax = R ∪ {−∞}. Among them is the
famous box-ball system [21], represented by the equation
U t+1
n = min
[
1− U tn,
n−1∑
k=−∞
U tk −
n−1∑
k=−∞
U t+1
k
]
.
Defining Stn =
∞∑
k=n
t∑
l=−∞
U lk, we obtain
St+1
n+1 + St−1n = max
[
St−1n+1 + St+1
n − 1, Stn + Stn+1
]
, (1.1)
which is ultradiscretization of the discrete KdV equation
(1 + δ)σt+1
n+1σ
t−1
n = δσt−1n+1σ
t+1
n + σtnσ
t
n+1. (1.2)
mailto:incidence_algebra@poset.jp
http://dx.doi.org/10.3842/SIGMA.2013.068
2 K. Kondo
Ultradiscretization [22] is a systematic procedure to obtain ultradiscrete systems from discrete
systems. The fundamental formula of the procedure is
lim
ε→+0
ε log
(
eA/ε + eB/ε
)
= max[A,B], lim
ε→+0
ε log
(
eA/s · eB/s
)
= A+B.
This may be understood as a transformation of addition into max operation and of multiplication
into addition. Setting δ = e−1/ε in (1.2) and applying ultradiscretization, we obtain (1.1).
The problem is, however, that ultradiscretization cannot be applied to subtraction, which is
of course necessary in many discrete integrable systems. The reason is as follows. If one wants
to define an ultradiscrete version of subtraction, it is natural to solve the linear equation
max[x, a] = b
for x ∈ Rmax. This has no solution when a > b, and therefore subtraction cannot be defined in
general.
Several attempts [8, 11, 15, 18, 19, 23] have been made to solve this problem. We focus on the
symmetrized max-plus algebra [1, 2], denoted by uR in this paper. This algebra is an extension
of Rmax and looks natural in the sense it traces the construction of Z from N2. Linear algebra
over uR is also possible, and ultradiscretization with uR is presented in [4]. These theories of
uR are mainly developed in the field of discrete event systems and seems little known to the
field of integrable systems.
1.2 Contents of the paper
The discrete sine-Gordon equation [3, 7]
(1− δ)τml τm+1
l+1 = τml+1τ
m+1
l − δσml+1σ
m+1
l ,
(1− δ)σml σm+1
l+1 = σml+1σ
m+1
l − δτml+1τ
m+1
l
has not been ultradiscretized until recent years because soliton solutions include subtraction or
even complex numbers. The first attempt is made by Isojima et al. [9, 10] where a τ -only trilinear
equation is exploited to exclude subtraction. Here we propose another method to ultradiscretize
the sine-Gordon equation which utilizes uR. The equation and the solutions are ultradiscretized
keeping subtraction and complex numbers in a highly direct fashion.
Noncommutative integrable systems have been drawing more interest in the last two decades.
It is difficult to point out the first appearance of such systems, but the noncommutative KdV
equation is already mentioned in [13]. The first discrete noncommutative integrable system
is probably the noncommutative discrete KP equation [12, 17]. Along this line, we propose
a noncommutative discrete analogue of the sine-Gordon equation, explore relations to other
integrable systems, and construct multisoliton solutions by the Darboux transformation. More-
over, we also propose a noncommutative ultradiscrete analogue of the sine-Gordon equation and
explicitly derive 1-soliton and 2-soliton solutions by ultradiscretization with uR. As a result, we
have a complete set of commutative and noncommutative versions of continuous, discrete, and
ultradiscrete sine-Gordon equations.
The rest of the paper is organized as follows. In Section 2.1, the discrete sine-Gordon equation
and 1-soliton and 2-soliton solutions are reviewed. Special solutions such as the traveling-
wave and kink-antikink solutions are presented probably for the first time. In Section 2.2, an
ultradiscrete analogue of the sine-Gordon equation is proposed and the solutions are obtained.
Because of ultradiscretization with uR, correspondence between the discrete and ultradiscrete
systems are direct, which is also supported by figures.
In Section 3.1, a noncommutative discrete analogue of the sine-Gordon equation is proposed.
A relation to other integrable systems including the noncommutative discrete KP equation is
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 3
explained, and multisoliton solutions are constructed by a repeated application of Darboux
transformations. In Section 3.2, a noncommutative ultradiscrete analogue of the sine-Gordon
equation is proposed and 1-soliton and 2-soliton solutions are derived. Also figures of solutions
for both equations are displayed.
In Section 4, concluding remarks and discussions are presented.
2 Discrete and ultradiscrete sine-Gordon equations
In this section, we first review the discrete sine-Gordon equation [3, 7] and several results around
it. Explicit calculation of the traveling-wave, kink-antikink, kink-kink, and breather solutions
are probably presented for the first time.
Then, we propose an ultradiscrete analogue of the sine-Gordon equation. The solutions are
obtained in two ways: by calculations completely inside uR, and by ultradiscretization with uR.
The correspondence between the discrete and ultradiscrete systems is quite clear. Similarity
of profiles of solutions is also visually confirmed by figures. Our formulation is different from
Isojima et al. [9].
2.1 Discrete sine-Gordon equation
We review the three representations of the discrete sine-Gordon equation, their connection to
the sine-Gordon equation, and some solutions for the sine-Gordon equation.
For any function f = f(l,m) over Z2, define shift operations by
fl = fl(l,m) = f(l + 1,m), fm = fm(l,m) = f(l,m+ 1).
Inverse operations are denoted by
fl = f(l − 1,m), fm = f(l,m− 1).
Let τ = τ(l,m), σ = σ(l,m) be functions Z2 → C. Date et al. [3] gave the discrete sine-Gordon
equation (dsG) in the following form
(1− δ)ττlm = τlτm − δσlσm, (2.1a)
(1− δ)σσlm = σlσm − δτlτm, (2.1b)
where δ ∈ C× is a parameter with a small absolute value. The vacuum solution
τ = σ = 1
is the simplest solution, other than the null solution τ = σ = 0. Calculating the cross product
of the both sides of (2.1a), (2.1b), we have
ττlm(σlσm − δτlτm) = σσlm(τlτm − δσlσm) ⇐⇒
τlmσm
σlmτm
− τlσ
σlτ
+ δ
(
σmσ
τmτ
− τlmτl
σlmσl
)
= 0
and thus
wlm
wm
− wl
w
+ δ
(
1
wmw
− wlmwl
)
= 0, (2.2)
where w is defined by
w =
τ
σ
.
4 K. Kondo
If we introduce u defined by
u =
2
i
logw,
we have
ei(ulm−um)/2 − ei(ul−u)/2 + δ
(
ei(−um−u)/2 − ei(ulm+ul)/2
)
= 0
⇐⇒ sin
(
ulm − ul − um + u
4
)
= δ sin
(
ulm + ul + um + u
4
)
. (2.3)
Each of (2.2) and (2.3) is also called the discrete sine-Gordon equation, where (2.3) is the original
form discovered by Hirota [7].
Remark 2.1. By the non-autonomous transformation w′ = w(−1)m , (2.2) becomes
w′m
w′lm
−
w′l
w′
+ δ
(
w′m
w′
−
w′l
w′lm
)
= 0 ⇐⇒
w′m − δw′l
w′lm
=
w′l − δw′m
w′
.
This is known as a discrete analogue of the modified KdV equation [16], and its ultradiscretiza-
tion is also known [20].
Assume u is also a function u(x, y) of continuum variables x, y ∈ R and has an expansion
u(x+ r, y + s) = u+ (rux + suy) +
1
2
(
r2uxx + 2rsuxy + s2uyy
)
+ · · · ,
where ux = ∂u/∂x, etc. Connect l, m to x, y via the Miwa transformation
u(x, y; l,m) = u(x+ la, y +mb)
where a, b ∈ R× are parameters. Then we have
ulm − ul − um + u = abuxy + (higher-order terms of a, b).
Setting δ = ab and taking the limit a, b→ 0 of (2.3) successively, we obtain
lim
a,b→0
1
ab
sin
(
ulm − ul − um + u
4
)
= lim
a,b→0
1
ab
sin
(
abuxy + · · ·
4
)
=
1
4
uxy,
lim
a,b→0
sin
(
ulm + ul + um + u
4
)
= sinu,
and thus the (continuous) sine-Gordon equation
uxy = 4 sinu.
This is known to have following special types of solutions (see, for example, [6]): the traveling-
wave solution
u = 4arctan exp
(
± x− vy√
1− v2
)
, (2.4)
the kink-antikink solution
u = 4arctan
(
sinh vy√
1−v2
v cosh x√
1−v2
)
, (2.5)
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 5
the kink-kink solution
u = 4arctan
(
v sinh x√
1−v2
cosh vy√
1−v2
)
, (2.6)
and the breather solution
u = 4arctan
(√
1− ω2
ω
sinωy
cosh
√
1− ω2 x
)
, (2.7)
where v, ω are constants.
2.1.1 1-soliton and 2-soliton solutions
Isojima et al. [9] have given the following conditions for τ and σ to be a 1-soliton or 2-soliton
solution. As a 1-soliton solution, assume
τ = 1 + f, σ = 1− f, f = cplqm (2.8)
where c, p, q ∈ C× are constants. By substitution, the dispersion relation
(1− δ)(1 + pq) = (1 + δ)(p+ q) ⇐⇒ q =
(1 + δ)p− (1− δ)
(1− δ)p− (1 + δ)
(2.9)
is found to be a necessary and sufficient condition. As a 2-soliton solution, assume
τ = 1 + f1 + f2 + αf1f2, σ = 1− f1 − f2 + αf1f2, fj = cjp
l
jq
m
j ,
where α ∈ C× is a constant. This time, the pair of the dispersion relation
(1− δ)(1 + pjqj) = (1 + δ)(pj + qj) ⇐⇒ qj =
(1 + δ)pj − (1− δ)
(1− δ)pj − (1 + δ)
and the relation
α = − p1 − p2
1− p1p2
q1 − q2
1− q1q2
=
(
p1 − p2
1− p1p2
)2
(2.10)
is a necessary and sufficient condition.
Fig. 2.1 shows the 1-soliton solution with
δ = 0.04, c = −1, p = 2,
and the 2-soliton solution with
δ = 0.04, c1 = c2 = −2.125, p1 = q2 = 2,
in the light-cone coordinates
(n, t) =
(
l +m
2
,
l −m
2
)
⇐⇒ (l,m) = (n+ t, n− t). (2.11)
6 K. Kondo
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
w
Figure 2.1. 1-soliton solution (left) and 2-soliton solution (right) for dsG.
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
Re w
-10
-5
0
5
10
n -10
-5
0
5
10
t
0
0.5
1
Im w
-10
-5
0
5
10
n -10
-5
0
5
10
t
0
2
4
6
u
Figure 2.2. Traveling-wave solution for dsG.
2.1.2 Traveling-wave solution
In Sections 2.1.2–2.1.4, we give solutions for (2.1a) and (2.1b) which correspond to the contin-
uous counterparts (2.4)–(2.7). These do not seem to be previously presented in the literature,
including Hirota [7] and Isojima et al. [9].
Replacing c by ic in the 1-soliton solution, we obtain
w =
1 + ic(pq)n
(
pq−1
)t
1− ic(pq)n
(
pq−1
)t , u = 4arctan
(
c(pq)n
(
pq−1
)t)
(2.12)
in the light-cone coordinates. This corresponds to the traveling-wave solution (2.4) for the
sine-Gordon equation (if we restrict δ, c, p ∈ R×). Fig. 2.2 shows the solution with
δ = 0.04, c = 1, p = 2.
2.1.3 Kink-antikink and kink-kink solutions
Set p1 = q2 in the 2-soliton solution. Then
p1 =
(1 + δ)p2 − (1− δ)
(1− δ)p2 − (1 + δ)
⇐⇒ p2 =
(1 + δ)p1 − (1− δ)
(1− δ)p1 − (1 + δ)
(2.13)
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 7
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
Re w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
Im w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-6
-4
-2
0
2
4
6
u
Figure 2.3. Kink-antikink solution for dsG.
and thus p2 = q1. Rewriting in the light-cone coordinates, we have
τ = 1 + c1p
n+t
1 pn−t2 + c2p
n+t
2 pn−t1 + αc1c2(p1p2)
2n
= (p1p2)
n
(
αc1c2(p1p2)
n + (p1p2)
−n + c1
(
p1p
−1
2
)t
+ c2
(
p1p
−1
2
)−t)
,
σ = (p1p2)
n
(
αc1c2(p1p2)
n + (p1p2)
−n − c1
(
p1p
−1
2
)t − c2 (p1p−12
)−t)
.
We set
β = ± p1 − p2
1− p1p2
, c1 = −c2 = iβ−1
and define
ch(p, l) =
pl + p−l
2
, sh(p, l) =
pl − p−l
2
.
Then
w =
β ch(p1p2, n) + i sh
(
p1p
−1
2 , t
)
β ch(p1p2, n)− i sh
(
p1p
−1
2 , t
) , u = 4arctan
(
sh
(
p1p
−1
2 , t
)
β ch(p1p2, n)
)
. (2.14)
This corresponds to the kink-antikink solution (2.5). Fig. 2.3 shows the solution with
δ = 0.04, c1 = −c2 = 2.125i, p1 = 2.
Similarly, setting p1q2 = 1 gives
p−11 =
(1 + δ)p2 − (1− δ)
(1− δ)p2 − (1 + δ)
⇐⇒ p−12 =
(1 + δ)p1 − (1− δ)
(1− δ)p1 − (1 + δ)
and thus p2q1 = 1. We have
τ = β−1(p1p2)
t
(
β(p1p2)
t + β(p1p2)
−t + i
(
p1p
−1
2
)n − i (p1p−12
)−n)
,
σ = β−1(p1p2)
t
(
β(p1p2)
t + β(p1p2)
−t − i−1
(
p1p
−1
2
)n
+ i
(
p1p
−1
2
)−n)
8 K. Kondo
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
Re w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
Im w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-6
-4
-2
0
2
4
6
u
Figure 2.4. Kink-kink solution for dsG.
for the same β, c1, c2 defined above and
w =
β ch(p1p2, t) + i sh
(
p1p
−1
2 , n
)
β ch(p1p2, t)− i sh
(
p1p
−1
2 , n
) , u = 4arctan
(
sh
(
p1p
−1
2 , n
)
β ch(p1p2, t)
)
. (2.15)
This corresponds to the kink-kink solution (2.6). Fig. 2.4 shows the solution with
δ = 0.04, c1 = −c2 = −0.470588i, p1 = 2.
2.1.4 Breather solution
Consider the kink-antikink solution where p1 and p2 are complex numbers satisfying
p1p2 ∈ R>0,
∣∣p1p−12
∣∣ = 1.
Such p1, p2 are complex conjugates of each other. If we write p1 = g + hi, p2 = g − hi and
substitute these into (2.13), we find g, h must satisfy
(1− δ)
(
1 + g2 + h2
)
= 2(1 + δ)g ⇐⇒ (1− δ)g2 − 2(1 + δ)g + (1− δ)
(
1 + h2
)
= 0.
As a quadratic equation of g, the condition for the existence of real roots is given by
(1 + δ)2 − (1− δ)2
(
1 + h2
)
≥ 0 ⇐⇒ h2 ≤
(
1 + δ
1− δ
)2
− 1.
Such a real number h does exist if δ ≥ 0, and g is given by
g =
1 + δ
1− δ
±
√(
1 + δ
1− δ
)2
− (1 + h2).
Rewriting p1 = reiθ, p2 = re−iθ, we obtain β = iγ where γ is defined by
γ = ±2r sin θ
1− r2
,
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 9
-10
-5
0
5
10
n -10
-5
0
5
10
t
0
0.5
1
Re w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
Im w
-10
-5
0
5
10
n -10
-5
0
5
10
t
-3
-2
-1
0
1
2
3
u
Figure 2.5. Breather solution for dsG.
and sh
(
p1p
−1
2 , t
)
= i sin 2tθ. Thus,
w =
γ ch(r2, n) + i sin 2tθ
γ ch(r2, n)− i sin 2tθ
, u = 4arctan
(
sin 2tθ
γ ch(r2, n)
)
.
This corresponds to the breather solution (2.7). Fig. 2.5 shows the solution with
δ = 0.04, c1 = −c2 = 0.75, p1 = 0.75 + 0.25i.
2.2 Ultradiscrete sine-Gordon equation
2.2.1 Ultradiscrete sine-Gordon equation
In order to ultradiscretize the discrete sine-Gordon equation (2.1a), (2.1b), we must deal with
negative numbers since either or both of τ and σ include subtractions. We adopt ultradiscretiza-
tion with the symmetrized max-plus algebra uR. For details, see Appendix A and references
cited there.
We perform ultradiscretization of dsG through the parametrization
δ = µDe
D̃s, D̃ < 0. (2.16)
This can be regarded as an other aspect of continuum limit since δ → 0 as s → ∞. Assuming
δ
ud−→ D, τ
ud−→ T , σ
ud−→ S, we obtain
TTlm ∇ TlTm DSlSm, (2.17a)
SSlm ∇ SlSm DTlTm. (2.17b)
We call the pair (2.17a), (2.17b) the ultradiscrete sine-Gordon equation (udsG). The vacuum
solution T = S = 0 is the simplest solution, other than the null solution T = S = −∞. We can
also ultradiscretize (2.2) to obtain
WlmW
−1
m WlW
−1 ⊕D
(
W−1m W−1 WlmWl
)
∇ −∞ (2.18)
where w
ud−→ W ∇ TS−1. We also call (2.18) the ultradiscrete sine-Gordon equation. The
ultradiscretization of (2.3) is unclear.
10 K. Kondo
2.2.2 Deterministic time evolution and class of solutions
It seems sensible to restrict ourselves to the class of signed solutions, that is, T, S,W ∈ uC∨ for
any (l,m) ∈ Z2 since it permits basic properties like weak substitution. The null and vacuum
solutions are signed solutions.
The problem is that udsG no longer admits time evolution, at least deterministic one, in
general, since the balance relation is not equality. For example, if we have
f(t+ 1) ∇
(
expression including f(t)
)
= 3•,
we cannot determine f(t+1) from f(t), since this relation is satisfied whenever |f(t+ 1)|⊕ ≤ 3.
Strictly speaking, udsG is not an equation.
But in some cases, it actually becomes an equation, or furthermore, a deterministically evo-
lutionary form. Multiplying T−1 to (2.17a) and S−1 to (2.17b), we have
T−1TTlm ∇ T−1 (TlTm DSlSm) , S−1SSlm ∇ S−1 (SlSm DTlTm) .
If T−1T = S−1S = 0 and the right hand sides are signed, we obtain
Tlm = T−1 (TlTm DSlSm) , (2.19a)
Slm = S−1 (SlSm DTlTm) (2.19b)
by reduction of balances (see Appendix A). We call (2.19a), (2.19b) the deterministically evolu-
tionary form of udsG. If we replace D by D and restrict ranges of D, T , S to R for example, the
assumptions are satisfied, and we obtain the completely ordinary-looking ultradiscrete equation:
Tlm = max(Tl + Tm, D + Sl + Sm)− T, (2.20a)
Slm = max(Sl + Sm, D + Tl + Tm)− S. (2.20b)
Deterministic time evolution is also possible in other settings, which are presented in the fol-
lowing sections.
It might be natural to think we should consider (2.19a), (2.19b), or even (2.20a), (2.20b)
only. However, it seems that the former cannot capture the traveling-wave, kink-antikink, and
kink-kink solutions. And the latter does not even seem to contain soliton solutions. Therefore,
we consider (2.17a), (2.17b) primarily.
We show one example of positive (R) time evolution by (2.20a), (2.20b) in Fig. 2.6. Initial
values are set as
T (l,−10) = l, S(−10,m) = m, −10 ≤ l,m ≤ 10
with D = −1. The sign (in the sense of ±) of the solution alternates with time.
2.2.3 1-soliton solution
Consider a signed solution T , S satisfying
T ∇ 0⊕ F, S ∇ 0 F, F = CP lQm, (2.21)
where C ∈ uC∨ and P,Q ∈ uR⊗. Weakly substituting these into (2.17a), (2.17b), we have
0⊕ PQF 2 ⊕ (0⊕ PQ)F ∇ 0⊕ PQF 2 ⊕ (P ⊕Q)F, (2.22a)
0⊕ PQF 2 (0⊕ PQ)F ∇ 0⊕ PQF 2 (P ⊕Q)F, (2.22b)
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 11
-10
-5
0
5
10
l -10
-5
0
5
10
m
-10
-5
0
5
10
Figure 2.6. Positive (R) evolution of udsG.
where 0⊕D = 0 D = 0 is used. The dispersion relation
0⊕ PQ ∇ P ⊕Q (2.23)
is a sufficient condition for (2.22a), (2.22b) to hold, since we can construct them by adding and
multiplying same numbers to the both sides of (2.23). Rewriting (2.23), we have
(P 0)Q ∇ (P 0)
and thus P = 0 or Q = 0. Obviously, (2.21) and (2.23) can be obtained by ultradiscretizing (2.8)
and (2.9), respectively, through
c = µCe
C̃s ud−→ C, p = µP e
P̃ s ud−→ P, q
ud−→ Q
or
c = µCe
C̃s ud−→ C, p
ud−→ P, q = µQe
Q̃s ud−→ Q.
The solution is, however, not completely determined yet, because the balance relation is not
equality as stated before. So we try to utilize reduction of balances. If C ∈ uZ is an odd number
and P,Q ∈ uZ are even numbers, then F is always odd and 0⊕F , 0 F can never be balanced
since 0 is even. By reduction of balances, we obtain
T = 0⊕ F, S = 0 F,
and W is also immediately determined since S−1 is signed. This solution admits deterministic
time evolution since
|TlTm|⊕ > |DSlSm|⊕ , |SlSm|⊕ > |DTlTm|⊕
and thus
T−1(TlTm DSlSm) = T−1TlTm ∈ uR⊗,
S−1(SlSm DTlTm) = S−1SlSm ∈ uR⊗.
Fig. 2.7 shows the solution with
D = −1, C = 1, P = 2
in the light-cone coordinates (2.11). It is somehow difficult to depict ultradiscrete numbers
in figures; here signs and absolute values are displayed separately, and signs are mapped from
0, 0•, ⊕0 to −1, 0, 1, respectively (balanced elements do not appear in the figure, though).
Observe that the form of the 1-soliton solution w for dsG is preserved in the signs. Absolute
values are always 0, corresponding to the fact that w asymptotically behaves as ±1.
12 K. Kondo
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs W
Figure 2.7. Signs (left) and absolute values (right) of 1-soliton solution for udsG.
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn uRe W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs uRe W
-10
-5
0
5
10
n -10
-5
0
5
10
t
0
1
2
sgn uIm W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-50
-40
-30
-20
-10
0
abs uIm W
Figure 2.8. Traveling-wave solution for udsG.
2.2.4 Traveling-wave solution
If we replace C by CI and redefine F = CP lQm (C ∈ uR⊗), we obtain
T = 0⊕ FI, S = 0 FI
and
W ∇
(
0 F 2
)
⊕ FI
0⊕ F 2
.
We choose odd C and even P , Q so that 0 F 2 is always signed and reduction of balances
can be applied. This solution no longer admits deterministic time evolution, but is apparently
ultradiscretization of the traveling-wave solution (2.12) for dsG. Fig. 2.8 shows the solution with
D = −1, C = 1, P = 2.
The uRe and uIm parts are displayed separately. The profile of the traveling-wave solution for
dsG is preserved well.
2.2.5 2-soliton solution
Assume
T ∇ 0⊕ F1 ⊕ F2 ⊕AF1F2, S ∇ 0 F1 F2 ⊕AF1F2, (2.24)
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 13
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs W
Figure 2.9. 2-soliton solution for udsG.
where A ∈ uR⊗ and Fj = CjP
l
jQ
m
j . We also assume
P1 6= P2, Q1 6= Q2.
By substitution, we find the pair of the dispersion relation
0⊕ PjQj ∇ Pj ⊕Qj
and the relation
A(0 P1P2)(0 Q1Q2)⊕ (P1 P2)(Q1 Q2) ∇ −∞ (2.25)
is a sufficient condition for (2.24) to become a solution. Obviously, (2.25) is ultradiscretization
of (2.10). When P1 = P2 = 0 or Q1 = Q2 = 0, any A satisfies (2.25). When P1 = Q2 = 0, we
have
A(0 P2)(0 Q1)⊕ (0 P2)(Q1 0) ∇ −∞ =⇒ A = 0.
The case P2 = Q1 = 0 is similar.
We can choose A,Cj , Pj , Qj ∈ uZ such that 0⊕ AF1F2 is always positive, even and F1 ⊕ F2
is negative, odd. Then the solution is determined as
T = 0⊕ F1 ⊕ F2 ⊕AF1F2, S = 0 F1 F2 ⊕AF1F2.
This admits deterministic time evolution, of course. Fig. 2.9 shows the solution with
D = −1, C1 = C2 = 1, P1 = Q2 = 4.
2.2.6 Kink-antikink and kink-kink solutions
If we replace C1 by C1I, C2 by C2I, and redefine Fj = CjP
l
jQ
m
j (Cj ∈ uR⊗) in the 2-soliton
solution, we obtain
T ∇ 0⊕AF1F2 ⊕ (F1 F2) I, S ∇ 0⊕AF1F2 (F1 F2) I
and
W ∇
(
(0⊕AF1F2)
2 (F1 F2)
2
)
⊕ (0⊕AF1F2)(F1 F2)I
(0⊕AF1F2)2 ⊕ (F1 F2)2
.
We choose Cj , Pj , Qj ∈ uZ such that
|F1|⊕ ≡ 1, |F2|⊕ ≡ 3 (mod 4)
14 K. Kondo
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn uRe W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs uRe W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn uIm W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-50
-40
-30
-20
-10
0
abs uIm W
Figure 2.10. Kink-antikink solution for udsG.
or
|F1|⊕ ≡ 3, |F2|⊕ ≡ 1 (mod 4).
Then F1 F2 and (0⊕AF1F2)
2 (F1 F2)
2 are always signed and the balance relations become
equalities.
If we set P1 = Q2, P2 = Q1, we have the kink-antikink solution. Similarly, setting P1 = Q−12 ,
P2 = Q−11 gives the kink-kink solution. These solutions does not admit deterministic time
evolution, but are ultradiscretization of (2.14), (2.15). The ultradiscretization of the breather
solution is unclear.
Fig. 2.10 shows the kink-antikink solution with
D = −1, C1 = 1, C2 = −1, P1 = Q2 = 4,
and Fig. 2.11 shows the kink-kink solution with
D = −1, C1 = 1, C2 = (−1), P1 = Q−12 = 4.
Observe that in the uIm part, two waves approach to each other for t < 0, collide at t = 0, and
move away from each other for t > 0. In the kink-antikink solution, the two waves have the
same sign and bump up by collision. In the kink-kink solution, the two have opposite signs and
reflect by collision.
3 Noncommutative discrete
and ultradiscrete sine-Gordon equations
In this section, we propose a noncommutative discrete analogue of the sine-Gordon equation as
a compatibility condition of a certain linear system. This equation reduces to the commutative
version once the underlying algebra turns out to be commutative and one simple reduction con-
dition is applied. A reduction from the noncommutative discrete KP equation [12, 17] also gives
the equation, and continuum limit of the equation gives the noncommutative (continuous) sine-
Gordon equation, which is already known in a different context [14]. We define the Darboux
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 15
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn uRe W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs uRe W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn uIm W
-10
-5
0
5
10
n -10
-5
0
5
10
t
-50
-40
-30
-20
-10
0
abs uIm W
Figure 2.11. Kink-kink solution for udsG.
transformation, which constructs new solutions from old ones, and obtain Casoratian-type solu-
tions by repeating it. Explicitly setting the starting solutions for repetition, we derive so-called
multisoliton solutions.
Along the construction of Casoratian-type solutions, quasideterminants [5] are used, which
is a noncommutative extension of determinants. The theory needs some space for explanation,
but it is not essential to the main story. Therefore, we only briefly explain the definition and
some properties of them in Appendix B. For details, see [5].
We finally propose a noncommutative ultradiscrete analogue of the sine-Gordon equation.
Noncommutative ultradiscrete setting is probably one of the hardest environments for integrable
systems to exist, but we manage to obtain 1-soliton and 2-soliton solutions by ultradiscretization.
Notations are slightly changed in this section because of the complexity of expressions we are
going to manipulate. Shifts are always indicated after a comma like f,l. This is to distinguish
indices and shifts. In addition, shift operators Tl, Tm are also used:
Tlf = f,l = f(l + 1,m), Tmf = f,m = f(l,m+ 1).
Do not confuse these with the ultradiscretized τ function of the previous section; in the non-
commutative setting, τ functions do not seem to exist. We also use superscripts for elements of
matrices. For example,
w = (wικ) =
w11 · · · w1N
...
. . .
...
wN1 · · · wNN
.
3.1 Noncommutative discrete sine-Gordon equation
3.1.1 Linear system
Let w = w(l,m), v = v(l,m) be functions Z2 → Mat(N,C) and
Bl =
(
w,lw
−1 −aλ
−aλ v,lv
−1
)
, Bm =
(
1 −bλ−1w,mv−1
−bλ−1v,mw−1 1
)
,
16 K. Kondo
where a, b, λ ∈ C× are parameters. Consider the linear system
Tl
(
φ
ψ
)
= Bl
(
φ
ψ
)
, Tm
(
φ
ψ
)
= Bm
(
φ
ψ
)
(3.1)
for φ, ψ : Z2 → Mat(N,C). Denoting entrywise shift operations by TmBl = Bl,m etc., we have
TmTl
(
φ
ψ
)
= Bl,mTm
(
φ
ψ
)
= Bl,mBm
(
φ
ψ
)
,
TlTm
(
φ
ψ
)
= Bm,lTl
(
φ
ψ
)
= Bm,lBl
(
φ
ψ
)
.
These must coincide, so we require the compatibility condition
Bl,mBm = Bm,lBl.
This is equivalent to
w,lmw
−1
,m − w,lw−1 + ab
(
v,mw
−1 − w,lmv−1,l
)
= 0, (3.2a)
v,lmv
−1
,m − v,lv−1 + ab
(
w,mv
−1 − v,lmw−1,l
)
= 0. (3.2b)
We call the pair (3.2a) and (3.2b) the noncommutative discrete sine-Gordon equation (ncdsG).
Proposition 3.1. When N = 1, the reduction condition
wv = 1 (3.3)
gives the (commutative) discrete sine-Gordon equation [3, 7]
w,lm
w,m
−
w,l
w
+ ab
(
1
w,mw
− w,lmw,l
)
= 0. (3.4)
Proof. Under (3.3), (3.2a) is apparently equivalent to (3.4). Since
(l.h.s. of (3.2a))×
(
w,mw −
1
w,lmw,l
)
= w,lmw +
1
w,lmw
− w,lw,m −
1
w,lw,m
= (l.h.s. of (3.2b))×
(
1
w,mw
− w,lmw,l
)
,
(3.2b) is also equivalent to (3.4). �
For any w0 satisfying
abw0,lm =
w0,lm
ab
,
(3.2a) and (3.2b) are solved by (w, v) =
(
w0, abw0,lm
)
, which is not an interesting solution. We
consider other types of solutions in the rest of this section.
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 17
3.1.2 Reduction from the noncommutative discrete KP equation
Let wi = wi(n1, n2, n3) (i = 1, 2, 3) be functions Z3 → Mat(N,C). The noncommutative discrete
KP equation [12, 17] is the set of equations
wi,j(ci − cj)w−1i + wj,k(cj − ck)w−1j + wk,i(ck − ci)w−1k = 0 (3.5)
for any combination of i, j, k ∈ {1, 2, 3}. Here i, j, k can take same values, and shifts are denoted
like
w1,2 = w1(n1, n2 + 1, n3).
ci ∈ C× are parameters taking mutually different values.
We replace (n1, n2, n3) by new coordinates (n′1, n
′
2, n
′
3) = (n1 − n3, n2, n3). Shifts are also in
new coordinates, and double-shifts are to be used:
w1,2 = w1(n
′
1, n
′
2 + 1, n′3), w2,13 = w2(n
′
1 + 1, n′2, n
′
3 + 1), etc.
Then, setting
δ =
c1 − c3
c1 − c2
,
we can rewrite (3.5) as
w1,2w
−1
1 + (δ − 1)w2,13w
−1
2 − δw3,1w
−1
3 = 0,
w1,2w
−1
1 = w2,1w
−1
2 , w2,13w
−1
2 = w3,2w
−1
3 , w3,1w
−1
3 = w1,13w
−1
1 .
By imposing the reduction condition
wi(n
′
1 + 2, n′2, n
′
3) = wi(n
′
1, n
′
2, n
′
3)
and defining vi = wi,1, we obtain
w1,2w
−1
1 + (δ − 1)v2,3w
−1
2 − δv3w
−1
3 = 0, (3.6a)
v1,2v
−1
1 + (δ − 1)w2,3v
−1
2 − δw3v
−1
3 = 0, (3.6b)
w1,2w
−1
1 = v2w
−1
2 , (3.6c)
v2,3w
−1
2 = w3,2w
−1
3 , (3.6d)
v3w
−1
3 = v1,3w
−1
1 , (3.6e)
v1,2v
−1
1 = w2v
−1
2 , (3.6f)
w2,3v
−1
2 = v3,2v
−1
3 , (3.6g)
w3v
−1
3 = w1,3v
−1
1 . (3.6h)
Proposition 3.2. For any w1, v1 satisfying (3.6a)–(3.6h),
(w, v) = (w1, v1) (l = n′2,m = n′3)
solves (3.2a) and (3.2b) with ab = δ.
Proof. Let us rewrite (3.6a) using only w1, v1. From (3.6e) we immediately have
w1,2w
−1
1 + (δ − 1)v2,3w
−1
2 − δv1,3w
−1
1 = 0, (3.7)
and thus try to rewrite the second term.
18 K. Kondo
(3.6d) implies
(δ − 1)v2,3w
−1
2 = (δ − 1)w3,2w
−1
3 = w3,2v
−1
3,2 · (δ − 1)v3,2v
−1
3 · v3w
−1
3 . (3.8)
Then, (3.6h) implies
w3,2v
−1
3,2 = w1,23v
−1
1,2, v3w
−1
3 = v1w
−1
1,3, (3.9)
and (3.6g) implies
(δ − 1)v3,2v
−1
3 = (δ − 1)w2,3v
−1
2 .
By (3.6b) and (3.6h), this equation becomes
(δ − 1)w2,3v
−1
2 = δw3v
−1
3 − v1,2v
−1
1 = δw1,3v
−1
1 − v1,2v
−1
1 . (3.10)
Combining (3.8)–(3.10), we obtain
(δ − 1)v2,3w
−1
2 = w1,23v
−1
1,2 ·
(
δw1,3v
−1
1 − v1,2v
−1
1
)
· v1w−11,3 = δw1,23v
−1
1,2 − w1,23w
−1
1,3.
Finally, (3.7) becomes
w1,2w
−1
1 − w1,23w
−1
1,3 + δ
(
w1,23v
−1
1,2 − v1,3w
−1
1
)
= 0. (3.11a)
In the same way, (3.6b) becomes
v1,2v
−1
1 − v1,23v
−1
1,3 + δ
(
v1,23w
−1
1,2 − w1,3v
−1
1
)
= 0. (3.11b)
If we set
(w, v) = (w1, v1), (l,m) = (n′2, n
′
3), ab = δ,
we have
w1,2 = w,l, w1,23 = w,lm, etc.
Therefore, (3.11a) and (3.11b) become (3.2a) and (3.2b). �
Remark 3.3. The solution constructed here seems to be only a part of the whole solutions
of (3.2a) and (3.2b), since it satisfies extra conditions
w1,2w
−1
1 · v1,2v
−1
1 = 1, v1,3w
−1
1 · w1,3v
−1
1 = 1.
3.1.3 Continuum limit
Assume w is also a function w(x, t) of continuum variables x, t ∈ R and has an expansion
w(x+ r, t+ s) = w + (rwx + swt) +
1
2
(
r2wxx + 2rswxt + s2wtt
)
+ · · · ,
where wx = ∂w/∂x, etc. Connect l, m to x, t via the Miwa transformation
w(x, t; l,m) = w(x+ la, t+mb).
Assume similarly for v = v(x, t; l,m). Then we have
w,l = w + awx +
a2
2
wxx + · · · ,
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 19
w,lm = w + (awx + bwt) +
1
2
(
a2wxx + 2abwxt + b2wtt
)
+ · · · ,
w−1,m = w−1 − bw−1wtw−1 −
b2
2
(
w−1wttw
−1 − 2w−1wtw
−1wtw
−1)+ · · · ,
v,l = · · · ,
and from (3.2a), (3.2b)
0 = w,lmw
−1
,m − w,lw−1 + ab
(
v,mw
−1 − w,lmv−1,l
)
= ab
(
wxtw
−1 − wxw−1wtw−1 + vw−1 − wv−1
)
+ (higher-order terms),
0 = ab
(
vxtv
−1 − vxv−1vtv−1 + wv−1 − vw−1
)
+ (higher-order terms).
Taking the limit a, b→ 0 successively, we obtain
wxtw
−1 − wxw−1wtw−1 + vw−1 − wv−1 = 0,
vxtv
−1 − vxv−1vtv−1 + wv−1 − vw−1 = 0.
Since −w−1wtw−1 = (w−1)t, these are transformed into(
wxw
−1)
t
= wv−1 − vw−1, (3.12a)(
wxw
−1 + vxv
−1)
t
= 0. (3.12b)
We call the pair (3.12a) and (3.12b) the noncommutative sine-Gordon equation. A quite similar
equation with the same name has been derived in a different context [14, (3.10)].
Proposition 3.4. When N = 1, the reduction condition
wv = 1 (3.13)
gives the (commutative) sine-Gordon equation
uxt = 4 sinu, (3.14)
where u is defined by
u =
2
i
logw.
Proof. Under (3.13), (3.12b) clearly holds. And (3.14) is immediate from (3.12a) since
uxt =
2
i
wxtw − wxwt
w2
, sinu =
w2 − w−2
2i
. �
3.1.4 Darboux transformation
When (w, v) is a solution for (3.2a) and (3.2b), the column vector
t(
φ ψ
)
satisfying the linear
system (3.1) is called the eigenfunction of (w, v) for eigenvalue λ.
Let
t(
φλ ψλ
)
,
t(
φµ ψµ
)
be eigenfunctions of (w, v) for eigenvalues λ, µ, respectively. Define
the Darboux transformation of (w, v) and
t(
φλ ψλ
)
by
t(
φµ ψµ
)
as
w̃ = ψµφ
−1
µ w, ṽ = φµψ
−1
µ v,
(
φ̃λ
ψ̃λ
)
= K
(
φλ
ψλ
)
where
K =
(
−µψµφ−1µ λ
λ −µφµψ−1µ
)
.
20 K. Kondo
Theorem 3.5. (w̃, ṽ) is a solution to (3.2a) and (3.2b), and
t(
φ̃λ ψ̃λ
)
is an eigenfunction of
(w̃, ṽ) for eigenvalue λ.
Proof. From the linear system (3.1), we can write
w,lw
−1 = (φµ,l + aµψµ)φ
−1
µ , v,lv
−1 = (ψµ,l + aµφµ)ψ
−1
µ ,
w,mv
−1 = b−1µ(φµ − φµ,m)ψ−1µ , v,mw
−1 = b−1µ(ψµ − ψµ,m)φ−1µ .
Then we have
w̃,lw̃
−1 = ψµ,l
(
ψ−1µ + aµφ−1µ,l
)
= v,lv
−1 + aµ
(
ψµ,lφ
−1
µ,l − φµψ
−1
µ
)
,
ṽ,lṽ
−1 = φµ,l
(
φ−1µ + aµψµ,l
)
= w,lw
−1 + aµ
(
φµ,lψ
−1
µ,l − ψµφ
−1
µ
)
,
w̃,mṽ
−1 = b−1µψµ,m
(
φ−1µ,m − φ−1µ
)
= vmw
−1 + b−1µ
(
ψµ,mφ
−1
µ,m − ψµφ−1µ
)
,
ṽ,mw̃
−1 = b−1µφµ,m
(
ψ−1µ,m − ψ−1µ
)
= w,mv
−1 + b−1µ
(
φµ,mψ
−1
µ,m − φµψµ
)
,
which imply(
w̃,lw̃
−1)
,m
− w̃,lw̃−1 + ab
(
ṽ,mw̃
−1 −
(
w̃,mṽ
−1)
,l
)
= 0,(
ṽ,lṽ
−1)
,m
− ṽ,lṽ−1 + ab
(
w̃,mṽ
−1 −
(
ṽ,mw̃
−1)
,l
)
= 0.
Define B̃l, B̃m by
B̃l =
(
w̃,lw̃
−1 −aλ
−aλ ṽ,lṽ
−1
)
, B̃m =
(
1 −bλ−1w̃,mṽ−1
−bλ−1ṽ,mw̃−1 1
)
.
Then we have
K,lBl = B̃lK, K,mBm = B̃mK
and thus
Tl
(
K
(
φ
ψ
))
= K,lBl
(
φ
ψ
)
= B̃l
(
K
(
φ
ψ
))
,
Tm
(
K
(
φ
ψ
))
= K,mBm
(
φ
ψ
)
= B̃m
(
K
(
φ
ψ
))
. �
3.1.5 Multisoliton solutions
The simplest solution for (3.2a) and (3.2b) is the vacuum solution (w, v) = (1, 1). The linear
system (3.1) of the vacuum solution is
Tl
(
φ
ψ
)
=
(
1 −aλ
−aλ 1
)(
φ
ψ
)
, Tm
(
φ
ψ
)
=
(
1 −bλ−1
−bλ−1 1
)(
φ
ψ
)
,
which has two basic solutions(
φ
ψ
)
=
(
(1− aλ)l
(
1− bλ−1
)m
(1− aλ)l
(
1− bλ−1
)m) ,( (1 + aλ)l
(
1 + bλ−1
)m
−(1 + aλ)l
(
1 + bλ−1
)m) .
Let λk (k = 1, 2, . . .) be mutually different eigenvalues and define
φk = (1− aλk)l
(
1− bλ−1k
)m
+ (1 + aλk)
l
(
1 + bλ−1k
)m
ck,
ψk = (1− aλk)l
(
1− bλ−1k
)m − (1 + aλk)
l
(
1 + bλ−1k
)m
ck,
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-2
-1.5
-1
-0.5
0
0.5
1
w
11
-10
-5
0
5
10
n -10
-5
0
5
10
t
0
0.5
1
1.5
2
2.5
3
3.5
w
12
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
w
21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
-0.5
0
0.5
1
1.5
2
w
22
Figure 3.1. 1-soliton solution for ncdsG.
where ck ∈ Mat(N,C) are parameters introducing noncommutativity.
t(
φk ψk
)
is of course an
eigenfunction of the vacuum solution for eigenvalue λk. Repeating the Darboux transformation
by
t(
φk ψk
)
, we can construct multisoliton solutions.
A 1-soliton solution is given by
w = ψ1φ
−1
1 = (1− f1)(1 + f1)
−1, (3.15a)
v = φ1ψ
−1
1 = (1 + f1)(1− f1)−1, (3.15b)
where fk is defined by
fk =
(
1 + aλk
1− aλk
)l(1 + bλ−1k
1− bλ−1k
)m
ck.
As a concrete example, Fig. 3.1 shows the behavior of
w =
(
w11 w12
w21 w22
)
(N = 2)
with
a = b = 0.2, c1 =
(
2 −4
1 −1.5
)
, λ1 =
5
3
in the light-cone coordinates (2.11).
A 2-soliton solution is given by
w =
(
λ2φ2 − λ1φ1ψ−11 ψ2
) (
λ2ψ2 − λ1ψ1φ
−1
1 φ2
)−1
ψ1φ
−1
1
=
(
λ2φ2ψ
−1
2 − λ1φ1ψ
−1
1
) (
λ2φ1ψ
−1
1 − λ1φ2ψ
−1
2
)−1
=
(
λ2(1 + f2)(1− f2)−1 − λ1(1 + f1)(1− f1)−1
)
×
(
λ2(1 + f1)(1− f1)−1 − λ1(1 + f2)(1− f2)−1
)−1
, (3.16a)
v =
(
λ2(1− f2)(1 + f2)
−1 − λ1(1− f1)(1 + f1)
−1)
22 K. Kondo
-10
-5
0
5
10
n -10
-5
0
5
10
t
-3
-2
-1
0
1
2
3
w
11
-10
-5
0
5
10
n -10
-5
0
5
10
t
-3
-2
-1
0
1
2
3
w
12
-10
-5
0
5
10
n -10
-5
0
5
10
t
-6
-4
-2
0
2
4
6
w
21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-3
-2
-1
0
1
2
3
w
22
Figure 3.2. 2-soliton solution for ncdsG.
×
(
λ2(1− f1)(1 + f1)
−1 − λ1(1− f2)(1 + f2)
−1)−1 . (3.16b)
Fig. 3.2 shows the solution with
a = b = 0.2, c1 =
(
2.5 −0.8
2 1.8
)
, c2 =
(
1.5 1.2
−1 0.5
)
, λ1 = λ−12 =
5
3
.
3.1.6 Casoratian-type solutions
Let (w, v) be a solution for (3.2a) and (3.2b),
t(
φk ψk
)
be eigenfunctions of (w, v) for eigen-
values λk (k = 1, 2, . . .), where λk are mutually different. Define repetition of the Darboux
transformation by
w(n) = ψ(n)
n
(
φ(n)n
)−1
w(n−1),
v(n) = φ(n)n
(
ψ(n)
n
)−1
v(n−1),
φ
(n+1)
k = λkψ
(n)
k − λnψ(n)
n
(
φ(n)n
)−1
φ
(n)
k ,
ψ
(n+1)
k = λkφ
(n)
k − λnφ
(n)
n
(
ψ(n)
n
)−1
ψ
(n)
k
and
w(0) = w, v(0) = v, φ
(1)
k = φk, ψ
(1)
k = ψk.
For notational convenience, we introduce reduced shift operator T defined by
Tf(φ1, ψ1, φ2, ψ2, . . .) = f(λ1ψ1, λ1φ1, λ2ψ2, λ2φ2, . . .),
where f(x1, x2, . . .) is any rational function of noncommutative variables xj . For example, we
have
Tφk = λkψk, Tψk = λkφk, T 2φk = λ2kφk.
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 23
Lemma 3.6.
Tφ
(n+1)
k = λkψ
(n+1)
k , Tψ
(n+1)
k = λkφ
(n+1)
k .
Proof. We prove by induction. Obviously, φ
(n+1)
k , ψ
(n+1)
k are rational functions of φj , ψj .
Assume Tφ
(n)
k = λkψ
(n)
k , Tψ
(n)
k = λkφ
(n)
k for certain n. Then,
Tφ
(n+1)
k = λk
(
λkφ
(n)
k
)
− λn
(
λnφ
(n)
n
)(
λnψ
(n)
n
)−1
λkψ
(n)
k = λkψ
(n+1)
k .
Similarly, Tψ
(n+1)
k = λkφ
(n+1)
k . �
Theorem 3.7.
w(n) =
n∏
j=1
(
−λ−1j
)
·
∣∣∣∣∣∣∣∣∣∣∣
φ1 φ2 · · · φn 1
Tφ1 Tφ2 · · · Tφn 0
...
...
. . .
...
...
Tn−1φ1 Tn−1φ2 · · · Tn−1φn 0
Tnφ1 Tnφ2 · · · Tnφn 0
∣∣∣∣∣∣∣∣∣∣∣
w, (3.17a)
v(n) =
n∏
j=1
(
−λ−1j
)
·
∣∣∣∣∣∣∣∣∣∣∣
ψ1 ψ2 · · · ψn 1
Tψ1 Tψ2 · · · Tψn 0
...
...
. . .
...
...
Tn−1ψ1 Tn−1ψ2 · · · Tn−1ψn 0
Tnψ1 Tnψ2 · · · Tnψn 0
∣∣∣∣∣∣∣∣∣∣∣
v, (3.17b)
φ
(n+1)
k =
∣∣∣∣∣∣∣∣∣∣∣
φ1 φ2 · · · φn φk
Tφ1 Tφ2 · · · Tφn Tφk
...
...
. . .
...
...
Tn−1φ1 Tn−1φ2 · · · Tn−1φn Tn−1φk
Tnφ1 Tnφ2 · · · Tnφn Tnφk
∣∣∣∣∣∣∣∣∣∣∣
, (3.17c)
ψ
(n+1)
k =
∣∣∣∣∣∣∣∣∣∣∣
ψ1 ψ2 · · · ψn ψk
Tψ1 Tψ2 · · · Tψn Tψk
...
...
. . .
...
...
Tn−1ψ1 Tn−1ψ2 · · · Tn−1ψn Tn−1ψk
Tnψ1 Tnψ2 · · · Tnψn Tnψk
∣∣∣∣∣∣∣∣∣∣∣
. (3.17d)
Here, quasideterminants [5] are used (see Appendix B). When n = 0, (3.17a) and (3.17b) read
w(0) = 1 ·
∣∣ 1 ∣∣w, v(0) = 1 ·
∣∣ 1 ∣∣v,
respectively.
Before proceeding to the proof, we prepare the following lemma.
Lemma 3.8. Let C = (cij), C
′ = (c′ij) be n × n matrices where cij , c
′
ij ∈ Mat(N,C). Assume
cij = c′ij for 1 ≤ i ≤ n, 1 ≤ j ≤ n− 1. Then,
|C|−1p1n|C
′|p1n = |C|−1p2n|C
′|p2n. (3.18)
Proof. By the column homological relation (Proposition B.3), we have
|Cp2n|−1p1j |C|p1n = −|Cp1n|p2j |C|p2n, (3.19a)
|C ′p2n|−1p1j |C
′|p1n = −|C ′p1n|p2j |C ′|p2n. (3.19b)
By the assumption, we have Cp2n = C ′p2n, Cp1n = C ′p1n. Therefore, we obtain (3.18) by
multiplying the inverse of (3.19a) to (3.19b) from the left. �
24 K. Kondo
Proof of Theorem 3.7. We prove by induction. The case n = 0 is trivial.
Assume w(n−1), v(n−1), φ
(n)
k , ψ
(n)
k have the above expressions for certain n > 0. Then,
w(n) = λ−1n
(
Tφ(n)n
)(
φ(n)n
)−1
w(n−1)
= −
n∏
j=1
(
−λ−1j
)
·
∣∣∣∣∣∣∣
Tφ1 · · · Tφn
...
. . .
...
Tnφ1 · · · Tnφn
∣∣∣∣∣∣∣
∣∣∣∣∣∣∣
φ1 · · · φn
...
. . .
...
Tn−1φ1 · · · Tn−1φn
∣∣∣∣∣∣∣
−1
×
∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 1
Tφ1 · · · Tφn−1 0
...
. . .
...
...
Tn−1φ1 · · · Tn−1φn−1 0
∣∣∣∣∣∣∣∣∣w.
By Lemma 3.8, we obtain
∣∣∣∣∣∣∣
φ1 · · · φn
...
. . .
...
Tn−1φ1 · · · Tn−1φn
∣∣∣∣∣∣∣
−1
∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 1
Tφ1 · · · Tφn−1 0
...
. . .
...
...
Tn−1φ1 · · · Tn−1φn−1 0
∣∣∣∣∣∣∣∣∣
=
∣∣∣∣∣∣∣
φ1 · · · φn
...
. . .
...
Tn−1φ1 · · · Tn−1φn
∣∣∣∣∣∣∣
−1
∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 1
Tφ1 · · · Tφn−1 0
...
. . .
...
...
Tn−1φ1 · · · Tn−1φn−1 0
∣∣∣∣∣∣∣∣∣ .
We define an (n− 1)× (n− 1) matrix A0 by
A0 =
Tφ1 · · · Tφn−1
...
. . .
...
Tn−1φ1 · · · Tn−1φn−1
.
With A0, the above quasideterminants are rewritten as
∣∣∣∣∣∣∣
Tφ1 · · · Tφn
...
. . .
...
Tnφ1 · · · Tnφn
∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣∣
Tφn
A0
...
Tn−1φn
Tnφ1 · · · Tnφn−1 Tnφn
∣∣∣∣∣∣∣∣∣ ,
∣∣∣∣∣∣∣
φ1 · · · φn
...
. . .
...
Tn−1φ1 · · · Tn−1φn
∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 φn
Tφn
A0
...
Tn−1φn
∣∣∣∣∣∣∣∣∣ ,∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 1
Tφ1 · · · Tφn−1 0
...
. . .
...
...
Tn−1φ1 · · · Tn−1φn−1 0
∣∣∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 1
0
A0
...
0
∣∣∣∣∣∣∣∣∣ ,
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 25
and also, we have a trivial identity
0 =
∣∣∣∣∣∣∣∣∣
0
A0
...
0
Tnφ1 · · · Tnφn−1 0
∣∣∣∣∣∣∣∣∣ .
By the invariance under row and column permutations (Proposition B.2) and Sylvester’s identity
(Proposition B.4), we can combine these four quasideterminants into one to obtain
w(n) =
n∏
j=1
(
−λ−1j
)
·
∣∣∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 φn 1
Tφn 0
A0
...
...
Tn−1φn 0
Tnφ1 · · · Tnφn−1 Tnφn 0
∣∣∣∣∣∣∣∣∣∣∣
w.
Similarly for v(n).
For φ
(n+1)
k , we have
φ
(n+1)
k = Tφ
(n)
k −
(
Tφ(n)n
)(
φ(n)n
)−1
φ
(n)
k
=
∣∣∣∣∣∣∣
Tφ1 · · · Tφn−1 Tφk
...
. . .
...
...
Tnφ1 · · · Tnφn−1 Tnφk
∣∣∣∣∣∣∣−
∣∣∣∣∣∣∣
Tφ1 · · · Tφn−1 Tφn
...
. . .
...
...
Tnφ1 · · · Tnφn−1 Tnφn
∣∣∣∣∣∣∣
×
∣∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 φn
Tφ1 · · · Tφn−1 Tφn
...
. . .
...
...
Tn−1φ1 · · · Tn−1φn−1 Tn−1φn
∣∣∣∣∣∣∣∣∣∣
−1
×
∣∣∣∣∣∣∣∣∣∣
φ1 · · · φn−1 φk
Tφ1 · · · Tφn−1 Tφk
...
. . .
...
...
Tn−1φ1 · · · Tn−1φn−1 Tn−1φk
∣∣∣∣∣∣∣∣∣∣
.
With the same technique for w(n), we obtain
φ
(n+1)
k =
∣∣∣∣∣∣∣
φ1 · · · φn φk
...
. . .
...
...
Tnφ1 · · · Tnφn Tnφk
∣∣∣∣∣∣∣ .
Similarly for ψ
(n+1)
k . �
3.2 Noncommutative ultradiscrete sine-Gordon equation
3.2.1 Ultradiscretization
We perform ultradiscretization of ncdsG by the parametrization
a = µAe
Ãs, b = µBe
Ãs, Ã, B̃ < 0.
26 K. Kondo
Assuming
a
ud−→ A, b
ud−→ B, w
ud−→W, v
ud−→ V,
we obtain
W,lmW
−1
,m W,lW
−1 ⊕AB
(
V,mW
−1 W,lmV
−1
,l
)
∇ −∞, (3.20a)
V,lmV
−1
,m V,lV −1 ⊕AB
(
W,mV
−1 V,lmW−1,l
)
∇ −∞. (3.20b)
We call the pair (3.20a) and (3.20b) the noncommutative ultradiscrete sine-Gordon equation
(ncudsG). Because uMat(N, uC) can be realized by uMat(2N, uR), we use uMat(N, uR) as the
underlying algebra for simplicity.
3.2.2 1-soliton solution
In order to ultradiscretize solutions for ncdsG, we introduce
pj =
1 + aλj
1− aλj
, qj =
1 + bλ−1j
1− bλ−1j
. (3.21)
These solve the dispersion relation
(1− ab)(1 + pjqj) = (1 + ab)(pj + qj), (3.22)
and any solution of (3.22) is parametrized by λj through (3.21) unless ab = 1. As in the
commutative case, (3.22) is ultradiscretized to
0⊕ PjQj ∇ Pj ⊕Qj , (3.23)
where pj
ud−→ Pj , qj
ud−→ Qj .
We can directly discretize the 1-soliton solution (3.15a), (3.15b) to obtain
w
ud−→W ∇ (0 F1)(0⊕ F1)
−1, v
ud−→ V ∇ (0⊕ F1)(0 F1)
−1,
where
Fj = P ljQ
m
j Cj , cj
ud−→ Cj ∈ uMat(N, uR).
This relation is valid, but inadequate to determine W , V in many cases. For simplicity, we
assume N = 2 hereafter. If we write W =
(
W ικ
)
, Fj =
(
F ικj
)
, the (1, 2)-th element of (0
F1)(0⊕ F1)
−1 is given by
F 12
1
((
0 F 11
1
)
⊕
(
0⊕ F 11
1
))
det(0⊕ F1)
=
F 12
1
(
0⊕
(
F 11
1
)•)
det(0⊕ F1)
,
and
∣∣F 11
1
∣∣
⊕ exceeds 0 for large ±l or ±m. Then this element is balanced and W 12 cannot be
determined. Therefore, we need more precise expressions to ultradiscretize.
Define
gj = (1 + fj)(1− fj)−1, hj = (1− fj)(1 + fj)
−1.
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 27
Of course, w = h1, v = g1 is a 1-soliton solution for ncdsG. Writing fj =
(
f ικj
)
, we have
gj =
(
1 + f11j
)(
1− f22j
)
+ f12j f
21
j
det(1− fj)
2f12j
det(1− fj)
2f21j
det(1− fj)
(
1 + f22j
)(
1− f11j
)
+ f21j f
12
j
det(1− fj)
,
hj =
(
1− f11j
)(
1 + f22j
)
+ f12j f
21
j
det(1 + fj)
−2f12j
det(1 + fj)
−2f21j
det(1 + fj)
(
1− f22j
)(
1 + f11j
)
+ f21j f
12
j
det(1 + fj)
.
By ultradiscretization, we obtain
gj
ud−→ Gj ∇
(
0⊕ F 11
j
)(
0 F 22
j
)
⊕ F 12
j F 21
j
det(0 Fj)
F 12
j
det(0 Fj)
F 21
j
det(0 Fj)
(0⊕ F 22
j )(0 F 11
j )⊕ F 21
j F 12
j
det(0 Fj)
, (3.24a)
hj
ud−→ Hj ∇
(
0 F 11
j
)(
0⊕ F 22
j
)
⊕ F 12
j F 21
j
det(0⊕ Fj)
F 12
j
det(0⊕ Fj)
F 21
j
det(0⊕ Fj)
(0 F 22
j )(0⊕ F 11
j )⊕ F 21
j F 12
j
det(0⊕ Fj)
. (3.24b)
We can choose Cj , Pj , Qj ∈ uZ such that
(
0 ⊕ F 11
j
)(
0 F 22
j
)
is always even and F 12
j F 21
j odd.
Then all the elements on the r.h.s. of (3.24a), (3.24b) are signed and Gj , Hj are completely
determined. Fig. 3.3 shows W = H1 with
A = B = −1, C1 =
(
(−7) −8
(−5) 7
)
, P1 = 2, Q1 = 0.
3.2.3 2-soliton solution
Ultradiscretization of (3.16a), (3.16b) gives
W ∇ (L2G2 L1G1) (L2G1 L1G2)
−1 ,
V ∇ (L2H2 L1H1) (L2H1 L1H2)
−1 ,
where
λj
ud−→ Lj .
In order to determine the value of Lj , we examine the relation
λj =
pj − 1
a(pj + 1)
=
b(qj + 1)
qj − 1
.
28 K. Kondo
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
11
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs W
11
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
12
-10
-5
0
5
10
n -10
-5
0
5
10
t
-50
-40
-30
-20
-10
abs W
12
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-50
-40
-30
-20
-10
abs W
21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
22
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs W
22
Figure 3.3. 1-soliton solution for ncudsG.
By the dispersion relation (3.23), we have Pj = 0 or Qj = 0. When Qj = 0, qj behaves like
a constant with regard to the ultradiscretization parameter s and pj cannot behave like one.
Therefore, we have
Lj =
Pj 0
A(Pj ⊕ 0)
=
{
A−1, |Pj |⊕ > 0,
A−1, |Pj |⊕ < 0.
Similarly, when Pj = 0, we have
Lj =
B(Qj ⊕ 0)
Qj 0
=
{
B, |Qj |⊕ > 0,
B, |Qj |⊕ < 0.
If we choose P2 = Q1 = 0, we have |L1|⊕ > |L2|⊕ and thus
W ∇ G1G
−1
2 , V ∇ H1H
−1
2 .
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 29
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
11
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs W
11
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
12
-10
-5
0
5
10
n -10
-5
0
5
10
t
-70
-60
-50
-40
-30
-20
-10
abs W
12
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-50
-40
-30
-20
-10
0
abs W
21
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
sgn W
22
-10
-5
0
5
10
n -10
-5
0
5
10
t
-1
0
1
abs W
22
Figure 3.4. 2-soliton solution for ncudsG.
Similarly, if P1 = Q2 = 0,
W ∇ G2G
−1
1 , V ∇ H2H
−1
1 .
Fig. 3.4 shows behavior of W = G1G
−1
2 with parameters A = B = −1,
C1 =
(
2 (−13)
11 15
)
, C2 =
(
−3 (−15)
(−7) (−13)
)
, P1 = Q2 = 4.
These are chosen so that every elements involved are signed.
4 Conclusion and discussion
We have proposed an ultradiscrete analogue of the sine-Gordon equation and constructed signed
1-soliton and 2-soliton solutions utilizing uR. The traveling-wave, kink-antikink, and kink-kink
30 K. Kondo
solutions, which contain ultradiscrete complex numbers, do exist and their correspondence to
those for the discrete sine-Gordon equation is quite clear. When the range of solutions are
restricted to uR, even deterministic time evolution is possible.
As stated in Section 1, another ultradiscretization of the sine-Gordon equation has been
given by Isojima et al. [9, 10]. There, only τ is ultradiscretized, no complex solutions are
dealt with, and time evolution is not possible. Our formulation looks better in these respects.
Also, ultradiscretization via the parametrization (2.16) can be considered as another aspect of
continuum limit, but ultradiscretization in [9] cannot since they choose a parametrization such
that δ → ±1.
We have also proposed a noncommutative discrete analogue of the sine-Gordon equation and
revealed its relation to other integrable systems including the noncommutative discrete KP equa-
tion. Also, multisoliton solutions are constructed by a repeated application of Darboux trans-
formations. And finally, a noncommutative ultradiscrete analogue of the sine-Gordon equation
and its signed 1-soliton and 2-soliton solutions are derived by ultradiscretization with uR.
A Symmetrized max-plus algebra and ultradiscretization
We make extensive use of the symmetrized max-plus algebra uR in the main part of the paper.
Therefore we describe basic definitions and properties of uR here. For details, see Baccelli et
al. [2].
A.1 Symmetrized max-plus algebra
A.1.1 Pair of the max-plus algebra
Let Rmax = R ∪ {−∞}. Rmax has the obvious total order. Define ⊕ and ⊗ by
x⊕ y = max(x, y), x⊗ y = x+ y
for x, y ∈ Rmax. With these operations, Rmax becomes a commutative dioid called the max-plus
algebra. The null element is −∞ and the unit element is 0. We extend ⊕ and ⊗ over R2
max by
(x1, x2)⊕ (y1, y2) = (x1 ⊕ y1, x2 ⊕ y2),
(x1, x2)⊗ (y1, y2) = (x1y1 ⊕ x2y2, x1y2 ⊕ x2y1).
Then R2
max is a commutative dioid with null element (−∞,−∞) and unit element (0,−∞).
Rmax is embedded into R2
max by x 7−→ (x,−∞).
Define minus sign by
(x1, x2) = (x2, x1)
for x = (x1, x2) ∈ R2
max. We write x y for x⊕ ( y), which is regarded as subtraction. Define
absolute value | |⊕ : R2
max → Rmax by
|(x1, x2)|⊕ = x1 ⊕ x2.
Define balance operator • by
(x1, x2)
• = (x1, x2) (x1, x2) = (x1 ⊕ x2, x1 ⊕ x2).
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 31
A.1.2 Symmetrized max-plus algebra
It is natural to consider the balance relation ∇ defined by
(x1, x2) ∇ (y1, y2) ⇐⇒ x1 ⊕ y2 = x2 ⊕ y1.
∇ is reflexive and symmetric, but not transitive. Therefore, we introduce another relation R
defined by
(x1, x2) R (y1, y2) ⇐⇒
{
(x1, x2) ∇ (y1, y2), when x1 6= x2 and y1 6= y2,
(x1, x2) = (y1, y2), otherwise.
R is an equivalence relation compatible with the operations ⊕, ⊗, , | |⊕, •, and the relation ∇.
Thus, we can define the quotient structure
uR = R2
max/R .
This is called the symmetrized max-plus algebra [1, 2]. Usually this is denoted by S, but we
use uR to imply it is somehow a whole set of ultradiscrete real numbers. We will also introduce
uZ,uC later.
Proposition A.1. We have three kinds of equivalence classes:
(x,−∞) = {(x, t) : t ∈ Rmax and x > t},
(−∞, x) = {(t, x) : t ∈ Rmax and t < x},
(x, x) = {(x, x)}.
Rmax is embedded into uR by x 7−→ (x,−∞). Define
Rmax =
{
(−∞, x) : x ∈ Rmax
}
, R•max =
{
(x, x) : x ∈ Rmax
}
.
Then uR has a decomposition
uR = Rmax ∪ Rmax ∪ R•max,
and (−∞,−∞) is the only element which belongs to any two of the three sets. Thus, we simply
write x for (x,−∞), x for (−∞, x), and x• for (x, x).
Define sign function sgnx by
sgnx =
0, x ∈ R,
0•, x ∈ R•max,
0, x ∈ R.
x ∈ uR is said to be positive if sgnx = 0, negative if sgnx = 0, and balanced if sgnx = 0•.
Define uR∨ = Rmax ∪ Rmax. x ∈ uR is said to be signed if x ∈ uR∨.
Proposition A.2. Let uR⊗ denote the whole set of invertible elements in uR. Then,
uR⊗ = uR∨ \ {−∞} = uR \ R•max.
Define uZ,uZ∨ ⊂ uR by
uZ = {−∞} ∪ Z ∪ Z ∪ Z•, uZ∨ = uZ ∩ uR∨
with obvious notations. uZ is a subdioid of uR and can be regarded as a whole set of ultradiscrete
integers. x ∈ uZ is said to be even if |x|⊕ is even, odd if |x|⊕ is odd. We do not define whether
−∞ is even or odd. We have of course
uZ⊗ = uZ∨ \ {−∞}.
32 K. Kondo
A.1.3 Properties of balance relation
We make much use of ∇, rather than R, since members of R•max can be regarded as a kind of
null elements by virtue of the following proposition.
Proposition A.3. For any x ∈ uR,
x ∇ −∞ ⇐⇒ x ∈ R•max.
Proposition A.4. For any x ∈ uR and t ∈ Rmax,
x ∇ t• and x 6∈ R•max ⇐⇒ |x|⊕ ≤ t.
Proposition A.5. For any x, y ∈ uR, we have
x ∇ y ⇐⇒ x y ∇ −∞.
Proposition A.6. For any x, y, z, w ∈ uR, we have
x ∇ y and z ∇ w =⇒ x⊕ z ∇ y ⊕ w,
x ∇ y =⇒ xz ∇ yz.
Proposition A.7 (weak substitution).
x ∇ y, cy ∇ z, and y ∈ uR∨ =⇒ cx ∇ z.
Corollary A.8 (weak transitivity).
x ∇ y, y ∇ z, and y ∈ uR∨ =⇒ x ∇ z.
Proposition A.9 (reduction of balances).
x ∇ y and x, y ∈ uR∨ =⇒ x = y.
A.1.4 Matrices and determinants
Let uMat(N, uR) denote the whole set of N ×N matrices over uR. Define addition ⊕ by
(aij)⊕ (bij) = (aij ⊕ bij)
and multiplication ⊗ by
(aij)⊗ (bij) = (cij), cij =
⊕
k
aik ⊗ bkj
for any (aij), (bij) ∈ uMat(N, uR). Then uMat(N, uR) becomes a dioid, noncommutative when
N > 1. , •, and ∇ are of course defined by
(aij) = ( aij), (aij)
• = (a•ij),
(aij) ∇ (bij) ⇐⇒ aij ∇ bij for any i, j
respectively. (aij) ∈ uMat(N, uR) is said to be signed if all the elements are signed. The
whole set of signed elements in uMat(N, uR) is denoted by uMat(N, uR)∨. uR is embedded into
uMat(N, uR) by
x 7−→
x −∞ · · · −∞
−∞ x
. . .
...
...
. . .
. . . −∞
−∞ · · · −∞ x
.
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 33
For any permutation σ ∈ SN , define sgn(σ) by
sgn(σ) =
{
0, when σ is even,
0, when σ is odd.
And define the determinant of a matrix A = (aij) ∈ uMat(N, uR) by
detA =
⊕
σ
sgn(σ)
⊗
i
aiσ(i).
detA is also denoted by |A| or |aij |.
Proposition A.10.
|tA| = |A|
where tA denotes transposition of A.
Proposition A.11.∣∣v1 · · · λvj ⊕ u · · · vN
∣∣ = λ
∣∣v1 · · · vj · · · vN
∣∣⊕ ∣∣v1 · · · u · · · vN
∣∣
where vj =
t(a1j , . . . , aNj) and u = t(u1, . . . , uN ).
Proposition A.12. For any permutation σ ∈ SN ,
|aiσ(j)| = sgn(σ)|aij |.
Corollary A.13. If vj = vk for some j 6= k, then∣∣v1 · · · vN
∣∣ ∇ −∞.
Let cofij(A) denote the cofactor of aij in |A|, which by definition satisfies
|A| =
⊕
i
aij ⊗ cofij(A)
for any j. Define the adjacent matrix of A by
adjA = (bij), bij = cofji(A).
Theorem A.14.
A⊗ adjA ∇ |A|, adjA⊗A ∇ |A|.
If |A| ∈ uR⊗, define A−1 by
A−1 = |A|−1 adjA.
This is not a multiplicative inverse in general, but plays a similar role with regard to∇. Therefore
we use the notation A−1.
34 K. Kondo
A.1.5 Ultradiscrete complex numbers
It is well known that we can construct complex numbers by 2× 2 real matrices, using
i =
(
0 −1
1 0
)
as the imaginary unit. Here we try to construct ultradiscrete complex numbers in a similar way.
Let uMat(N, uR) denote the algebra of N × N matrices whose elements are in uR. Define
I ∈ uMat(2,uR) by
I =
(
−∞ 0
0 −∞
)
.
We have
I2 =
(
−∞ 0
0 −∞
)(
−∞ 0
0 −∞
)
=
(
0 −∞
−∞ 0
)
= 0.
Define uC ⊂ uMat(2,uR) by
uC = {x⊕ yI | x, y ∈ uR}.
Proposition A.15. uC is a commutative subdioid of uMat(2, uR).
Proof. Obviously uC includes −∞ and 0. For any a⊕ bI, c⊕ dI ∈ uC, we have
(a⊕ bI)⊕ (c⊕ dI) = (a⊕ c)⊕ (b⊕ d)I ∈ uC,
(a⊕ bI)⊗ (c⊕ dI) = (ac bd)⊕ (ad⊕ bc)I ∈ uC.
And uC is commutative because I0 and I1 are commutative. �
When z ∈ uC is expressed as z = x+ yI where x, y ∈ uR, we write
uRe z = x, uIm = y.
The whole set of signed elements of uC is denoted by uC∨.
If det(x⊕ yI) = x2 ⊕ y2 ∈ uR⊗, we have
(x⊕ yI)−1 = x yI
x2 ⊕ y2
and
(x⊕ yI)(x⊕ yI)−1 = 0⊕ (xy)•
x2 ⊕ y2
I ∇ 0.
A.2 Ultradiscretization with negative numbers
Ultradiscretization with negative numbers is presented in De Schutter et al. [4]. Here we refor-
mulate it in a similar, but more convenient form for our purpose.
Let f(s) and g(s) be real functions. We say f(s) is asymptotically equivalent to g(s) if there
exists a real number s0 such that g(s) 6= 0 for any s > s0 and
lim
s→∞
f(s)
g(s)
= 1.
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 35
We also say f(s) is asymptotically equivalent to 0 if there exists a real number s1 such that
f(s) = 0 for any s > s1. Asymptotic equivalence is an equivalence relation and denoted by
f(s) ∼ g(s).
We are interested in asymptotic equivalence to exponential functions. If
f(s) ∼ µF eF̃ s, µF ∈ R×, F̃ ∈ R,
we write
f(s)
ud−→ F, F = S(µF )⊗ F̃ ∈ uR∨,
where
S(µ) =
{
0, µ > 0,
0, µ < 0.
We regard 0 ∼ µe(−∞)s for some µ ∈ R× and 0
ud−→ −∞ as a convention.
It is very important here to notice that µF is not restricted to positive numbers, unlike the
usual ultradiscretization procedure.
Proposition A.16 (ultradiscretization of addition). Let f(s), g1(s), . . . , gn(s) be real functions
satisfying
f(s) =
n∑
k=1
gk(s)
and
f(s)
ud−→ F, gk(s)
ud−→ Gk.
Then,
F ∇
n⊕
k=1
Gk.
Remark A.17. f(s)
ud−→ F and g(s)
ud−→ G do not imply f(s) + g(s)
ud−→ F ⊕ G because
f(s) + g(s) might be no longer asymptotically equivalent to exponential functions. But if f(s),
g(s) can be expressed by power series in δ = µDe
D̃s where D̃ < 0, this is not a problem because
f(s) + g(s) can also be expressed by a power series in δ.
Proposition A.18 (ultradiscretization of multiplication). Let f(s), g(s), h(s) be real functions
satisfying
f(s) = g(s)h(s)
and
g(s)
ud−→ G, h(s)
ud−→ H.
Then,
f(s)
ud−→ F = G⊗H.
36 K. Kondo
Corollary A.19 (ultradiscretization of polynomials). Let real functions f(s), gkl(s) satisfy
f(s) =
n∑
k=1
mk∏
l=1
gkl(s)
and
f(s)
ud−→ F, gkl(s)
ud−→ Gkl.
Then,
F ∇
n⊕
k=1
mk⊗
l=1
Gkl.
A.3 Ultradiscretization of matrices and complex numbers
We also reformulate ultradiscretization of matrices in [4]. Extension to complex numbers is
straightforward.
Consider a matrix-valued function f(s) = (fij(s)) : R→ Mat(N,R). If
fij(s)
ud−→ Fij ,
we write
f(s)
ud−→ F = (Fij) ∈ uMat(N, uR).
This is a componentwise property; there is no exponential functions of matrices.
Proposition A.20. Let matrix-valued functions f(s), gkl(s) satisfy
f(s) =
n∑
k=1
mk∏
l=1
gkl(s)
and
f(s)
ud−→ F, gkl(s)
ud−→ Gkl.
Then,
F ∇
n⊕
k=1
mk⊗
l=1
Gkl.
Considering 2× 2-matrix construction of complex numbers, we have
i =
(
0 −1
1 0
)
ud−→ I =
(
−∞ 0
0 −∞
)
.
Let f(s) = u(s) + v(s)i where u(s), v(s) are real functions. If
u(s)
ud−→ U, v(s)
ud−→ V,
we have of course
f(s)
ud−→ F = U ⊕ V I ∈ uC.
Ultradiscrete and Noncommutative Discrete sine-Gordon Equations 37
Proposition A.21. Let complex-valued functions f(s), gkl(s) satisfy
f(s) =
n∑
k=1
mk∏
l=1
gkl(s)
and
f(s)
ud−→ F, gkl(s)
ud−→ Gkl.
Then,
F ∇
n⊕
k=1
mk⊗
l=1
Gkl.
B Quasideterminants
Quasideterminants [5] are noncommutative extension of determinants, or, more precisely, deter-
minants divided by cofactors. Here we describe the definition and some properties required for
Theorem 3.7. See [5] for more detail.
Let R be a ring and Mat(N,R) be the whole set of N × N matrices over R. R is not
commutative in general. For any (aij), (bij) ∈ Mat(N,R), define addition by
(aij) + (bij) = (aij + bij)
and multiplication by
(aij)(bij) = (cij), cij =
N∑
k=1
aikbkj .
Ordering of multiplication is important here.
For any A = (aij) ∈ Mat(N,R), define the (p, q)-th quasideterminant |A|pq by
|A|pq = apq − rqp (Apq)
−1 cpq ,
where rqp is the p-th row of A without the q-th element, cpq is the q-th column of A without the
p-th element, and Apq is A without the p-th row and the q-th column. |A|pq is also written as
|A|pq =
∣∣∣∣∣∣∣
a11 · · · a1N
... apq
...
aN1 · · · aNN
∣∣∣∣∣∣∣ .
For example, we have∣∣∣∣ a11 a12
a21 a22
∣∣∣∣ = a11 − a12a−122 a21,
∣∣∣∣a11 a12
a21 a22
∣∣∣∣ = a12 − a11a−121 a22
and ∣∣∣∣∣∣
a11 a12 a13
a21 a22 a23
a31 a32 a33
∣∣∣∣∣∣ = a11 −
(
a12 a13
)(a22 a23
a32 a33
)−1(
a21
a31
)
= a11 − a12
(
a22 − a23a−133 a32
)−1
a21 − a13
(
a23 − a22a−132 a33
)−1
a21
− a12
(
a32 − a33a−123 a22
)−1
a31 − a13
(
a33 − a32a−122 a23
)−1
a31.
38 K. Kondo
Proposition B.1. If we write A−1 = (bij), we have
bij = |A|−1ji .
Proposition B.2. Quasideterminants are invariant under row and column permutations. (If
the row or column contains the box, it is moved together.)
Proposition B.3 (homological relations). For p1 6= p2, q1 6= q2, i 6= p, j 6= q, we have the row
homological relation
|A|pq1 |Apq2 |
−1
iq1
+ |A|pq2 |Apq1 |
−1
iq2
= 0
and the column homological relation
|Ap2q|−1p1j |A|p1q + |A
p1q|−1p2j |A|p2q = 0.
Proposition B.4 (Sylvester’s identity). For any A = (aij) ∈ Mat(N,R), define (N−k)×(N−k)
matrix A0 by
A0 = (aij), k + 1 ≤ i, j ≤ N
and k × k matrix C by
C = (cij), cij =
∣∣∣∣∣∣∣∣∣
aij ai(k+1) · · · aiN
a(k+1)j
... A0
aNj
∣∣∣∣∣∣∣∣∣ .
Then
|A|pq = |C|pq.
Acknowledgements
The author would like to express his gratitude to Professor Tetsuji Tokihiro, who provided
precise advices with a fine prospect. The author is also grateful to Professor Ralph Willox,
who provided helpful comments for refining the results. In addition, the author thanks the
anonymous referees for carefully reading the paper and giving many suggestions.
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1 Introduction
1.1 Ultradiscretization and its problem
1.2 Contents of the paper
2 Discrete and ultradiscrete sine-Gordon equations
2.1 Discrete sine-Gordon equation
2.1.1 1-soliton and 2-soliton solutions
2.1.2 Traveling-wave solution
2.1.3 Kink-antikink and kink-kink solutions
2.1.4 Breather solution
2.2 Ultradiscrete sine-Gordon equation
2.2.1 Ultradiscrete sine-Gordon equation
2.2.2 Deterministic time evolution and class of solutions
2.2.3 1-soliton solution
2.2.4 Traveling-wave solution
2.2.5 2-soliton solution
2.2.6 Kink-antikink and kink-kink solutions
3 Noncommutative discrete and ultradiscrete sine-Gordon equations
3.1 Noncommutative discrete sine-Gordon equation
3.1.1 Linear system
3.1.2 Reduction from the noncommutative discrete KP equation
3.1.3 Continuum limit
3.1.4 Darboux transformation
3.1.5 Multisoliton solutions
3.1.6 Casoratian-type solutions
3.2 Noncommutative ultradiscrete sine-Gordon equation
3.2.1 Ultradiscretization
3.2.2 1-soliton solution
3.2.3 2-soliton solution
4 Conclusion and discussion
A Symmetrized max-plus algebra and ultradiscretization
A.1 Symmetrized max-plus algebra
A.1.1 Pair of the max-plus algebra
A.1.2 Symmetrized max-plus algebra
A.1.3 Properties of balance relation
A.1.4 Matrices and determinants
A.1.5 Ultradiscrete complex numbers
A.2 Ultradiscretization with negative numbers
A.3 Ultradiscretization of matrices and complex numbers
B Quasideterminants
References
|