Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property
The lattice sine-Gordon equation is an integrable partial difference equation on ℤ², which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property, as well as its autonomou...
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| description | The lattice sine-Gordon equation is an integrable partial difference equation on ℤ², which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property, as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 032, 8 pages
Properties of the Non-Autonomous
Lattice Sine-Gordon Equation:
Consistency around a Broken Cube Property
Nobutaka NAKAZONO
Institute of Engineering, Tokyo University of Agriculture and Technology,
2-24-16 Nakacho Koganei, Tokyo 184-8588, Japan
E-mail: nakazono@go.tuat.ac.jp
URL: https://researchmap.jp/nakazono/
Received February 03, 2022, in final form April 14, 2022; Published online April 20, 2022
https://doi.org/10.3842/SIGMA.2022.032
Abstract. The lattice sine-Gordon equation is an integrable partial difference equation
on Z2, which approaches the sine-Gordon equation in a continuum limit. In this paper,
we show that the non-autonomous lattice sine-Gordon equation has the consistency around
a broken cube property as well as its autonomous version. Moreover, we construct two new
Lax pairs of the non-autonomous case by using the consistency property.
Key words: lattice sine-Gordon equation; Lax pair; integrable systems; partial difference
equations
2020 Mathematics Subject Classification: 37K10; 39A14; 39A45
1 Introduction
The sine-Gordon equation:
ϕtt − ϕxx + sinϕ = 0, (1.1)
where ϕ = ϕ(t, x) ∈ C and (t, x) ∈ C2, is well known as a motion equation of a row of pendulums
hang from a rod and are coupled by torsion springs. This equation is also known as a famous
example of integrable systems. In 1992, the following autonomous difference equation was
found [3, 20]:
ul+1,m+1
ul,m
=
(
γ − ul+1,m
1− γul+1,m
)(
1− γul,m+1
γ − ul,m+1
)
, (1.2)
where ul,m ∈ C, (l,m) ∈ Z2 and γ ∈ C, which is a discrete analogue of equation (1.1) and
therefore called lattice sine-Gordon (lsG) equation. Note that it is not only equation (1.2) that is
named lattice/discrete sine-Gordon equation. Moreover, in 2018, the following non-autonomous
version of the lsG equation was found [11]:
ul+1,m+1
ul,m
=
(
pl+1 − qmul+1,m
qm − pl+1ul+1,m
)(
qm+1 − plul,m+1
pl − qm+1ul,m+1
)
, (1.3)
where pl, qm ∈ C are respectively arbitrary functions of l and m.
A Lax pair is known as one of the most famous and important objects in the theory of
integrable systems, which implies the integrability of differential/difference equations. A Lax
pair of equation (1.2) is already known [3, 8, 9, 20], but that of equation (1.3) has not yet been
reported.
mailto:nakazono@go.tuat.ac.jp
https://researchmap.jp/nakazono/
https://doi.org/10.3842/SIGMA.2022.032
2 N. Nakazono
In this paper, we focus on equation (1.3) and give its Lax pairs by using the consistency around
a broken cube (CABC) property. (See Appendix A for the definition of CABC property). The
motivation for the discovery of the Lax pairs of equation (1.3) is as follows:
� the autonomous lsG equation (1.2) has the CABC property [9];
� by using the CABC property, a Lax pair of equation (1.2) was constructed in [9].
From the facts above, we can expect that the non-autonomous lsG equation (1.3) also has the
CABC property and by using it a Lax pair of equation (1.3), as well as equation (1.2), can be
constructed. This prediction is correct, and these results are summarized in Section 1.1.
1.1 Main results
In this subsection, we present the main results of this paper.
Firstly, in Section 2, we shall give proofs of the following theorem.
Theorem 1.1. Equation (1.3) has the CABC property.
See Appendix A for the definition of CABC property. Using the CABC property, we obtain
the following theorems.
Theorem 1.2. The following system for the two-vector Φl,m:
Φl+1,m = Ll,mΦl,m, Φl,m+1 = Ml,mΦl,m,
with
Ll,m =
plpl+1
(
pl − qmul,m
qm − plul,m
)
ul+1,m − pl
2 κ+ pl
2
p1pl+1
(
pl − qmul,m
qm − plul,m
)
ul+1,m 0
,
Ml,m =
pl
2 − qm
2
qm − plul,m
κ+ pl
2
plul,m
p1
(
pl − qmul,m
qm − plul,m
)
pl
ul,m
− qm
,
where κ ∈ C is a spectral parameter, is a Lax pair of equation (1.3), that is, the compatibility
condition
Ll,m+1Ml,m = Ml+1,mLl,m
gives equation (1.3).
The proof of Theorem 1.2 is given in Section 2.1.
Theorem 1.3. The following system for the two-vector Ψl,m:
Ψl+1,m = Ll,mΨl,m, Ψl,m+1 = Ml,mΨl,m,
with
Ll,m =
0 1
κ
(
pl − qmul,m
qm − plul,m
)
ul+1,m 0
, Ml,m =
0
1
ul,m
κ
(
pl − qmul,m
qm − plul,m
)
0
,
Properties of the Non-Autonomous Lattice Sine-Gordon Equation 3
where κ ∈ C is a spectral parameter, is a Lax pair of equation (1.3), that is, the compatibility
condition
Ll,m+1Ml,m = Ml+1,mLl,m
gives equation (1.3).
The proof of Theorem 1.3 is given in Section 2.2.
1.2 Notation and terminology
For conciseness in the remainder of the paper, we adopt the following notation for an arbitrary
function xl,m:
x = xl,m, x = xl+1,m, x̃ = xl,m+1, x̃ = xl+1,m+1, (1.4)
and extend the notation to other iterates of x as needed.
We write each lattice equation as the vanishing condition of a polynomial of four variables.
For example, the lsG equation (1.3) is given by Q
(
u, u, ũ, ũ
)
= 0, where
Q
(
u, u, ũ, ũ
)
= ũ(qm − pl+1u)
(
pl − qm+1ũ
)
− u(pl+1 − qmu)
(
qm+1 − plũ
)
.
(Where convenient, we also use lattice equations in their equivalent rational forms.) Note that,
for conciseness, we omit the dependence of the polynomial Q on parameters. We assume that
any parameters in the polynomial take generic values and that the corresponding polynomial is
irreducible.
Because of the association with a quadrilateral of Z2, a lattice equation relating four vertex
values is called a quad-equation. By a small abuse of terminology, we will also refer to the
corresponding function, whose vanishing condition gives the lattice equation, as a quad-equation.
Moreover, if the polynomial defining a quad-equation is quadratic in each variable, we especially
refer to it as a multi-quadratic quad-equation.
1.3 Outline of the paper
This paper is organized as follows. In Section 2, showing the lattice structures of equation (1.3)
in two ways, we give the proofs of Theorems 1.1–1.3. Some concluding remarks are given in
Section 3. Moreover, in Appendix A, we give the definition of consistency around a broken cube
property.
2 Lattice structures of the lsG equation (1.3)
In this section, showing two types of lattice structures of equation (1.3) we give the proofs of
Theorems 1.1–1.3. See Appendix A for the definition of CABC and tetrahedron properties.
2.1 CABC property of equation (1.3): I
We here start by defining the system of P∆Es:
A
(
u, u, ũ, ũ
)
=
ũ
u
−
(
pl+1 − qmu
qm − pl+1u
)(
qm+1 − plũ
pl − qm+1ũ
)
= 0, (2.1a)
S
(
u, u, ṽ, ṽ
)
=
1
ṽ
− pl
2
κ+ pl2
+
plpl+1u(pl+1 − qmu)(1− ṽ)
(κ+ pl2)(qm − pl+1u)
= 0, (2.1b)
4 N. Nakazono
B
(
u, v, ṽ
)
= ṽ − (κ+ pl
2)(qm − plu) + pl(pl
2 − qm
2)uv
pl(pl − qmu)(qm − plu+ pluv)
= 0, (2.1c)
C
(
u, u, v, v
)
= v − 1− (qm − plu)(κ+ pl
2 − pl
2v)
plpl+1(pl − qmu)uv
= 0, (2.1d)
where we have used the terminology given in equation (1.4) for ul,m and vl,m. Note that equa-
tion (2.1a) is exactly equivalent to equation (1.3).
It is straightforward to confirm that the system (2.1) has the CABC and tetrahedron prop-
erties. The tetrahedron equations K1 = K1
(
u, u, v, ṽ
)
and K2 = K2
(
u, u, v, ṽ
)
are given by
K1 =
pl+1
(
ṽ − 1
)
(pl+1 − qmu)
qm − pl+1u
−
pl
(
κ+ qm
2
)
(pl − qmu)v(
κ+ pl2
)
(qm − plu) + pl
(
pl2 − qm2
)
uv
+ qm = 0,
K2 =
κ+ pl
2
plṽ
−
(
κ+ qm
2
)
u
qm − pl+1u+ pl+1u v
− pl + qmu = 0.
Therefore, Theorem 1.1 holds.
Moreover, from the system (2.1) we obtain the following equation given only by the vari-
able vl,m:(
pl+1
2(1− v)
(
1− ṽ
)
v − pl
2(1− v)(1− ṽ)ṽ
)(
pl
2(1− v)(1− ṽ)v − pl+1
2(1− v)
(
1− ṽ
)
ṽ
)
− qm
2
(
κ+ pl
2 + pl+1
2
)(
vv − ṽ ṽ
)2 − qm
2
(
pl
2 − pl+1
2
)(
v ṽ − vṽ
)(
vv − ṽ ṽ
)
+ κ2
(
v − ṽ
)
(v − ṽ) + κpl
2(1− v)(1− ṽ)
(
vv − 2v ṽ + ṽ ṽ
)
+ κpl+1
2(1− v)
(
1− ṽ
)(
vv − 2vṽ + ṽ ṽ
)
+ κqm
2
(
v + v − ṽ − ṽ
)(
vv − ṽ ṽ
)
+ qm
2pl
2(1 + vṽ)
(
v − ṽ
)(
vv − ṽ ṽ
)
+ qm
2pl+1
2
(
1 + v ṽ
)
(v − ṽ)
(
vv − ṽ ṽ
)
= 0, (2.2)
which is assigned on the top face of each broken cube (see Figure 1). See the proof of Theorem 2
in [9] for details on how to derive a difference equation given only by the variable vl,m from
a system of P∆Es which has the CABC property. Note that the system of equations (2.1b)–
(2.1d) can also be regarded as a Bäcklund transformation from equation (2.1a) to equation (2.2).
Remark 2.1. The system (39) in [9], which implies the CABC property of equation (1.2), can
be obtained from the system (2.1) with the following specialization and transformation:
pl = γ, qm = 1, (ul,m, vl,m) 7→
(
ul,m, γ−1vl,m
)
.
Next, we construct the Lax pair in Theorem 1.2 through a method that parallels the well-
known method for constructing a Lax pair using the consistency around a cube (CAC) property
[4, 13, 21]. Substituting
vl,m =
Fl,m
Gl,m
,
into the equations (2.1c) and (2.1d) and separating the numerators and denominators of the
resulting equations, we obtain the following linear systems:
Fl+1,m = δ
(1)
l,m
((
plpl+1
(
pl − qmu
qm − plu
)
u− pl
2
)
Fl,m +
(
κ+ pl
2
)
Gl,m
)
, (2.3a)
Gl+1,m = δ
(1)
l,mp1pl+1
(
pl − qmu
qm − plu
)
uFl,m, (2.3b)
Fl,m+1 = δ
(2)
l,m
(
pl
2 − qm
2
qm − plu
Fl,m +
κ+ pl
2
plu
Gl,m
)
, (2.3c)
Properties of the Non-Autonomous Lattice Sine-Gordon Equation 5
Gl,m+1 = δ
(2)
l,m
(
p1
(
pl − qmu
qm − plu
)
Fl,m +
(pl
u
− qm
)
Gl,m
)
, (2.3d)
where δ
(1)
l,m and δ
(2)
l,m are arbitrary decoupling factors. Then, letting
Φl,m =
(
Fl,m
Gl,m
)
,
and taking
δ
(1)
l,m = 1, δ
(2)
l,m = 1,
from the equations (2.3) we obtain the Lax pair in Theorem 1.2.
2.2 CABC property of equation (1.3): II
In this subsection, we show another system of P∆Es which also gives the CABC property of
equation (1.3). The process for demonstrating the result is exactly the same as that in Section 2.1
and so, for conciseness, we omit detailed arguments.
The system of P∆Es
A
(
u, u, ũ, ũ
)
=
ũ
u
−
(
pl+1 − qmu
qm − pl+1u
)(
qm+1 − plũ
pl − qm+1ũ
)
= 0, (2.4a)
S
(
u, u, ṽ, ṽ
)
=
1
ṽ
− κ(pl+1 − qmu)uṽ
qm − pl+1u
= 0, (2.4b)
B
(
u, v, ṽ
)
= ṽ − qm − plu
κ(pl − qmu)uv
= 0, (2.4c)
C
(
u, u, v, v
)
= v − qm − plu
κ(pl − qmu)uv
= 0, (2.4d)
where equation (2.4a) is exactly equivalent to equation (1.3), has the CABC and tetrahedron
properties. The tetrahedron equations are given by
K1 =
ṽ
v
− (qm − pl+1u)(pl − qmu)
(pl+1 − qmu)(qm − plu)
= 0,
K2 =
u
u
− ṽ
v
= 0,
and the equation represented only by the variable vl,m is given by
vv + ṽṽ − pl(1 + κvṽ)v ṽ
pl+1
(
1 + κv ṽ
) −
pl+1
(
1 + κv ṽ
)
vṽ
pl(1 + κvṽ)
−
κqm
2
(
vv − ṽ ṽ
)2
plpl+1(1 + κvṽ)
(
1 + κv ṽ
) = 0.
Therefore, the system (2.4) gives another proof of Theorem 1.1.
Moreover, the Lax pair in Theorem 1.3 can be obtained by using the system (2.4) in the same
way as in Section 2.1.
3 Concluding remarks
In this paper, we have shown the CABC property of the non-autonomous lattice sine-Gordon
equation (1.3). Moreover, using the CABC property we have constructed two Lax pairs for
equation (1.3).
6 N. Nakazono
Equation (1.3) has properties similar to those of the Hirota’s dKdV equation [7, 10, 19]:
ul+1,m+1 − ul,m =
qm+1 − pl
ul,m+1
− qm − pl+1
ul+1,m
, (3.1)
according to our recent series of studies. The common properties are, for example, that they
have the CABC property shown in [9] and in this paper, and that they are not included in the
list of equations, which have the CAC property, in [1, 2, 5, 6]. Also, in a recent paper by the
author [12], it was found that equation (3.1) has a special solution called the discrete Painlevé
transcendent solution. In fact, equation (1.3) also has a special solution of the same type. This
result will be reported in a forthcoming publication. It is expected that there are many more
equations besides the equations (1.3) and (3.1) that have these common properties. We plan to
derive equations with such properties in a future project.
A Consistency around a broken cube property
In this appendix, we recall the definition of consistency around a broken cube (CABC) property.
We refer the reader to [9] for detailed information about this property.
We assign the following eight variables:
u0, u1, u2, u12, v0, v1, v2, v12 ∈ C,
to vertexes of the cube as shown in Figure 1. In contrast to the usual procedure assumed for
proving the consistency around a cube (CAC) property [1, 2, 5, 6, 14, 15, 16, 17, 18], we do
not assign a quad-equation to each face of the cube. Instead, we describe a system of equations
on the cube, which may (i) vary with each face; (ii) become a triangular equation, i.e., those
relating only three vertex values, on certain faces; and, (iii) involves vertices of a quadrilateral
given by an interior diagonal slice of the cube.
Three of the quad-equations occur on the bottom, front and back faces of the cube, while the
fourth one occurs in the interior of the cube as a diagonal slice. Each triangular domain occurs
as a half of the left or right face of the cube. See Figure 1. We will refer to this configuration
as a broken cube.
Figure 1. A cube with three quadrilateral faces labelled by A, C and C′, an interior diagonal quadrilateral
labelled by S and triangular domains labelled as B and B′. Note that primes denote domains on parallel
faces.
Correspondingly, we define polynomials of 4 variables A,S, C, C′ : C4 → C and those of 3
variables B,B′ : C3 → C, such that B and B′ written as functions of (x, y, z) satisfy
1) degx B ≥ 1, degy B = degz B = 1;
Properties of the Non-Autonomous Lattice Sine-Gordon Equation 7
2) the equation B = 0 can be solved for y and z, and each solution is a rational function of
the other two arguments.
With the labelling of vertices given in Figure 1, we denote the system of six corresponding
equations by
A(u0, u1, u2, u12) = 0, S(u0, u1, v2, v12) = 0, (A.1a)
B(u0, v0, v2) = 0, B′(u1, v1, v12) = 0, (A.1b)
C(u0, u1, v0, v1) = 0, C′(u2, u12, v2, v12) = 0. (A.1c)
The following definition describes how consistency holds for this system of equations.
Definition A.1 (CABC property). Let {u0, u1, u2, v0} be given initial values. Using equa-
tions (A.1), we can express the variable v12 as a rational function in terms of the initial values
in 3 ways. When the 3 results for v12 are equal, the system of equations (A.1) is said to be con-
sistent around a broken cube or to have the consistency around a broken cube (CABC) property.
In this case, we refer to the configuration of quadrilaterals and triangular domains associated
with the polynomials A, S, C, C′, B, B′ as a CABC cube.
Other equations arise from interrelationships between the above equations on the broken
cube. For example, an equation arises on the top face, parallel to A. It is also useful to note
equations that relate three vertices on a face to a vertex on the opposite face. The following
definition of such equations uses terminology analogous to existing ones in the literature on the
CAC property.
Definition A.2 (tetrahedron property). A CABC cube is said to have a tetrahedron property,
if there exist quad-equations K1 and K2 satisfying
K1(u0, u1, v0, v12) = 0, K2(u0, u1, v1, v2) = 0.
In this case, each of the equations K1 = 0 and K2 = 0 is referred to as a tetrahedron equation.
By interpreting each vertex value as an iterate of a function in an appropriate way, we
can interpret the above equations as P∆Es. In particular, we use the terminology given in
equation (1.4) for ul,m and vl,m to give the following definition of P∆Es.
Definition A.3 (CABC and tetrahedron properties for a system of P∆Es). Define the P∆Es
A
(
u, u, ũ, ũ
)
= 0, S
(
u, u, ṽ, ṽ
)
= 0, B
(
u, v, ṽ
)
= 0, C
(
u, u, v, v
)
= 0, (A.2)
which give the following equations around each elementary cubic cell in Z3:
A = A
(
u, u, ũ, ũ
)
= 0, S = S
(
u, u, ṽ, ṽ
)
= 0, (A.3a)
B = B
(
u, v, ṽ
)
= 0, B′ = B
(
u, v, ṽ
)
= 0, (A.3b)
C = C
(
u, u, v, v
)
= 0, C′ = C
(
ũ, ũ, ṽ, ṽ
)
= 0. (A.3c)
Then, the system (A.2) is said to have the CABC property if Definition A.1 holds for the
equations (A.3). We also transfer the definition of tetrahedron properties to P∆Es corresponding
to Kj , j = 1, 2, in the obvious way. Moreover, the P∆E
A
(
u, u, ũ, ũ
)
= 0
will be described as having the CABC property, if the system (A.2) has the CABC property.
Remark A.4. Note that equations (A.2) are not necessarily autonomous. They may contain
parameters that evolve with (l,m).
8 N. Nakazono
Acknowledgment
This research was supported by a JSPS KAKENHI Grant Number JP19K14559.
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1 Introduction
1.1 Main results
1.2 Notation and terminology
1.3 Outline of the paper
2 Lattice structures of the lsG equation (1.3)
2.1 CABC property of equation (1.3): I
2.2 CABC property of equation (1.3): II
3 Concluding remarks
A Consistency around a broken cube property
References
|
| id | nasplib_isofts_kiev_ua-123456789-211636 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T20:29:55Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nakazono, Nobutaka 2026-01-07T13:42:42Z 2022 Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property. Nobutaka Nakazono. SIGMA 18 (2022), 032, 8 pages 1815-0659 2020 Mathematics Subject Classification: 37K10; 39A14; 39A45 arXiv:2201.11264 https://nasplib.isofts.kiev.ua/handle/123456789/211636 https://doi.org/10.3842/SIGMA.2022.032 The lattice sine-Gordon equation is an integrable partial difference equation on ℤ², which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property, as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property. This research was supported by a JSPS KAKENHI Grant Number JP19K14559. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property Article published earlier |
| spellingShingle | Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property Nakazono, Nobutaka |
| title | Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property |
| title_full | Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property |
| title_fullStr | Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property |
| title_full_unstemmed | Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property |
| title_short | Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property |
| title_sort | properties of the non-autonomous lattice sine-gordon equation: consistency around a broken cube property |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211636 |
| work_keys_str_mv | AT nakazononobutaka propertiesofthenonautonomouslatticesinegordonequationconsistencyaroundabrokencubeproperty |