Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice
We recently introduced a class of ZN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called ''self-dual''. In this paper we discuss the continuous symmetries of these systems,...
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Fordy, A.P. Xenitidis, P. 2019-02-18T15:52:28Z 2019-02-18T15:52:28Z 2017 Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice / A.P. Fordy, P. Xenitidis // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 6 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K05; 37K10; 37K35; 39A05 DOI:10.3842/SIGMA.2017.051 https://nasplib.isofts.kiev.ua/handle/123456789/148563 We recently introduced a class of ZN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called ''self-dual''. In this paper we discuss the continuous symmetries of these systems, their reductions and the relation of the latter to the Bogoyavlensky equation. PX acknowledges support from the EPSRC grant Structure of partial difference equations with continuous symmetries and conservation laws, EP/I038675/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice Article published earlier |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice Fordy, A.P. Xenitidis, P. |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice |
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self-dual systems, their symmetries and reductions to the bogoyavlensky lattice |
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Fordy, A.P. Xenitidis, P. |
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Fordy, A.P. Xenitidis, P. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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We recently introduced a class of ZN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called ''self-dual''. In this paper we discuss the continuous symmetries of these systems, their reductions and the relation of the latter to the Bogoyavlensky equation.
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1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/148563 |
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Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice / A.P. Fordy, P. Xenitidis // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 6 назв. — англ. |
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2025-11-25T22:45:06Z |
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2025-11-25T22:45:06Z |
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1850570539863638016 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 051, 10 pages
Self-Dual Systems, their Symmetries
and Reductions to the Bogoyavlensky Lattice
Allan P. FORDY † and Pavlos XENITIDIS ‡
† School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
E-mail: A.P.Fordy@leeds.ac.uk
‡ School of Mathematics, Statistics and Actuarial Science, University of Kent,
Canterbury CT2 7FS, UK
E-mail: P.Xenitidis@kent.ac.uk
Received May 01, 2017, in final form June 26, 2017; Published online July 06, 2017
https://doi.org/10.3842/SIGMA.2017.051
Abstract. We recently introduced a class of ZN graded discrete Lax pairs and studied
the associated discrete integrable systems (lattice equations). In particular, we introduced
a subclass, which we called “self-dual”. In this paper we discuss the continuous symmetries of
these systems, their reductions and the relation of the latter to the Bogoyavlensky equation.
Key words: discrete integrable system; Lax pair; symmetry; Bogoyavlensky system
2010 Mathematics Subject Classification: 37K05; 37K10; 37K35; 39A05
1 Introduction
We recently introduced a class of ZN graded discrete Lax pairs and studied the associated
discrete integrable systems [2, 3]. Many well known examples belong to that scheme for N = 2,
so, for N ≥ 3, some of our systems may be regarded as generalisations of these.
In this paper we give a short review of our considerations and discuss the general framework
for the derivation of continuous flows compatible with our discrete Lax pairs. These derivations
lead to differential-difference equations which define generalised symmetries of our systems [2].
Here we are interested in the particular subclass of self-dual discrete integrable systems, which
exist only for N odd [3], and derive their lowest order generalised symmetries which are of
order one. We also derive corresponding master symmetries which allow us to construct infinite
hierarchies of symmetries of increasing orders.
These self-dual systems also have the interesting property that they can be reduced from
N − 1 to N−1
2 components, still with an N × N Lax pair. However not all symmetries of our
original systems are compatible with this reduction. From the infinite hierarchies of generalised
symmetries only the even indexed ones are reduced to corresponding symmetries of the reduced
systems. Thus the lowest order symmetries of our reduced systems are of order two.
Another interesting property of these differential-difference equations is that they can be
brought to a polynomial form through a Miura transformation. In the lowest dimensional case
(N = 3) this polynomial equation is directly related to the Bogoyavlensky equation (see (5.2)
below), whilst the higher dimensional cases can be regarded as multicomponent generalisations
of the Bogoyavlensky lattice.
Our paper is organised as follows. Section 2 contains a short review of our framework, the fully
discrete Lax pairs along with the corresponding systems of difference equations, and Section 3
discusses continuous flows and symmetries. The following section discusses the self-dual case
This paper is a contribution to the Special Issue on Symmetries and Integrability of Difference Equations.
The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html
mailto:A.P.Fordy@leeds.ac.uk
mailto:P.Xenitidis@kent.ac.uk
https://doi.org/10.3842/SIGMA.2017.051
http://www.emis.de/journals/SIGMA/SIDE12.html
2 A.P. Fordy and P. Xenitidis
and the reduction of these systems. It also presents the systems and corresponding reductions for
N = 3, 5 and 7. Section 5 presents the Miura transformations for the reduced systems in N = 3
and 5, and discusses the general formulation of these transformations for any dimension N .
2 ZN -graded Lax pairs
We now consider the specific discrete Lax pairs, which we introduced in [2, 3]. Consider a pair
of matrix equations of the form
Ψm+1,n = Lm,nΨm,n ≡
(
Um,n + λΩ`1
)
Ψm,n, (2.1a)
Ψm,n+1 = Mm,nΨm,n ≡
(
Vm,n + λΩ`2
)
Ψm,n, (2.1b)
where
Um,n = diag
(
u(0)m,n, . . . , u
(N−1)
m,n
)
Ωk1 , Vm,n = diag
(
v(0)m,n, . . . , v
(N−1)
m,n
)
Ωk2 , (2.1c)
and
(Ω)i,j = δj−i,1 + δi−j,N−1.
The matrix Ω defines a grading and the four matrices of (2.1) are said to be of respective le-
vels ki, `i, with `i 6= ki (for each i). The Lax pair is characterised by the quadruple (k1, `1; k2, `2),
which we refer to as the level structure of the system, and for consistency, we require
k1 + `2 ≡ k2 + `1 (modN).
Since matrices U , V and Ω are independent of λ, the compatibility condition of (2.1),
Lm,n+1Mm,n = Mm+1,nLm,n,
splits into the system
Um,n+1Vm,n = Vm+1,nUm,n, (2.2a)
Um,n+1Ω
`2 − Ω`2Um,n = Vm+1,nΩ`1 − Ω`1Vm,n, (2.2b)
which can be written explicitly as
u
(i)
m,n+1v
(i+k1)
m,n = v
(i)
m+1,nu
(i+k2)
m,n , (2.3a)
u
(i)
m,n+1 − u
(i+`2)
m,n = v
(i)
m+1,n − v
(i+`1)
m,n , (2.3b)
for i ∈ ZN .
2.1 Quotient potentials
Equations (2.3a) hold identically if we set
u(i)m,n = α
φ
(i)
m+1,n
φ
(i+k1)
m,n
, v(i)m,n = β
φ
(i)
m,n+1
φ
(i+k2)
m,n
, i ∈ ZN , (2.4)
after which (2.3b) takes the form
α
(
φ
(i)
m+1,n+1
φ
(i+k1)
m,n+1
−
φ
(i+`2)
m+1,n
φ
(i+`2+k1)
m,n
)
= β
(
φ
(i)
m+1,n+1
φ
(i+k2)
m+1,n
−
φ
(i+`1)
m,n+1
φ
(i+`1+k2)
m,n
)
, i ∈ ZN , (2.5)
Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice 3
defined on a square lattice. These equations can be explicitly solved for the variables on any of
the four vertices and, in particular,
φ
(i)
m+1,n+1 =
φ
(i+k1)
m,n+1φ
(i+k2)
m+1,n
φ
(i+k1+`2)
m,n
(
αφ
(i+`2)
m+1,n − βφ
(i+`1)
m,n+1
αφ
(i+k2)
m+1,n − βφ
(i+k1)
m,n+1
)
, i ∈ ZN . (2.6)
In this potential form, the Lax pair (2.1) can be written
Ψm+1,n =
(
αφm+1,nΩk1φ−1m,n + λΩ`1
)
Ψm,n,
Ψm,n+1 =
(
βφm,n+1Ω
k2φ−1m,n + λΩ`2
)
Ψm,n, (2.7a)
where
φm,n := diag
(
φ(0)m,n, . . . , φ
(N−1)
m,n
)
and det
(
φm,n
)
=
N−1∏
i=0
φ(i)m,n = 1. (2.7b)
We can then show that the Lax pair (2.7) is compatible if and only if the system (2.5) holds.
3 Differential-difference equations as symmetries
Here we briefly outline the construction of continuous isospectral flows of the Lax equations (2.1),
since these define continuous symmetries for the systems (2.3). The most important formula for
us is (3.4), which gives the explicit form of the symmetries in potential form.
We seek continuous time evolutions of the form
∂tΨm,n = Sm,nΨm,n,
which are compatible with each of the discrete shifts defined by (2.1), if
∂tLm,n = Sm+1,nLm,n − Lm,nSm,n,
∂tMm,n = Sm,n+1Mm,n −Mm,nSm,n.
Since
∂t(Lm,n+1Mm,n −Mm+1,nLm,n)
= Sm+1,n+1(Lm,n+1Mm,n −Mm+1,nLm,n)− (Lm,n+1Mm,n −Mm+1,nLm,n)Sm,n,
we have compatibility on solutions of the fully discrete system (2.2).
If we define Smn by
Sm,n = L−1m,nQm,n, where Qm,n = diag
(
q(0)m,n, q
(1)
m,n, . . . , q
(N−1)
m,n
)
Ωk1 , (3.1)
then
Qm,nUm−1,n − Um,nΩ−`1Qm,nΩ`1 = 0, ∂tUm,n = Ω−`1Qm+1,nΩ`1 −Qm,n,
which are written explicitly as
q(i)m,nu
(i+k1)
m−1,n = u(i)m,nq
(i+k1−`1)
m,n , (3.2a)
∂tu
(i)
m,n = q
(i−`1)
m+1,n − q
(i)
m,n. (3.2b)
It is also possible to prove (see [2]) that
N−1∑
i=0
q
(i)
m,n
u
(i)
m,n
=
1
αN
, (3.2c)
for an autonomous symmetry.
Equations (3.2a) and (3.2c), fully determine the functions q
(i)
m,n in terms of um,n and um−1,n.
4 A.P. Fordy and P. Xenitidis
Remark 3.1 (symmetries in m- and n-directions). The formula (3.1) defines a symmetry which
(before prolongation) only involves shifts in the m-direction. There is an analogous symmetry
in the n-direction, defined by
∂sΨm,n =
(
Vm,n + λΩ`2
)−1
Rm,nΨm,n, with Rm,n = diag
(
r(0)m,n, . . . , r
(N−1)
m,n
)
Ωk2 . (3.3)
Master symmetry and the hierarchy of symmetries
The vector field XM , defined by the evolution
∂τu
(i)
m,n = (m+ 1)q
(i−`)
m+1,n −mq
(i)
m,n, ∂τα = 1/
(
NαN−1
)
,
with q(i) being the solution of (3.2a) and (3.2c), is a master symmetry of X1, satisfying[[
XM , X1
]
, X1
]
= 0, with
[
XM , X1
]
6= 0.
We then define Xk recursively by Xk+1 = [XM , Xk]. We have
Proposition 3.2. Given the sequence of vector fields Xk, defined above, we suppose that, for
some ` ≥ 2, {X1, . . . , X`} pairwise commute. Then [Xi, X`+1] = 0, for 1 ≤ i ≤ `− 1.
This follows from an application of the Jacobi identity.
Remark 3.3. We cannot deduce that [[XM , X`], X`] = 0 by using the Jacobi identity. Since
we are given this equality for ` = 2, we can deduce that [X1, X3] = 0 (see the discussion around
Theorem 19 of [6]). Nevertheless it is possible to check this by hand for low values of `, for all
the examples given in this paper.
3.1 Symmetries in potentials variables
If we write (3.2) and the corresponding n-direction symmetry in the potential variables (2.4),
we obtain
∂tφ
(i)
m,n = α−1q(i−`1)m,n φ
(i+k1)
m−1,n −
φ
(i)
m,n
NαN
, (3.4a)
∂sφ
(i)
m,n = β−1q(i−`2)m,n φ
(i+k2)
m,n−1 −
φ
(i)
m,n
NβN
. (3.4b)
Remark 3.4. The symmetry (3.4a) is a combination of the “generalised symmetry” (3.2) and
a simple scaling symmetry, with coefficient chosen so that the vector field is tangent to the level
surfaces
N−1∏
i=0
φ
(i)
m,n = const, so this symmetry survives the reduction to N −1 components, which
we always make in our examples. The symmetry (3.4b) is similarly related to (3.3) and also
survives the reduction to N − 1 components.
The master symmetries are similarly adjusted, to give
∂τφ
(i)
m,n = mα−1q(i−`1)m,n φ
(i+k1)
m−1,n −
mφ
(i)
m,n
NαN
, (3.5a)
∂σφ
(i)
m,n = nβ−1q(i−`2)m,n φ
(i+k2)
m,n−1 −
nφ
(i)
m,n
NβN
, (3.5b)
where ∂τα = 1/
(
NαN−1
)
and ∂σβ = 1/
(
NβN−1
)
.
Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice 5
4 The self-dual case
In [3] we give a number of equivalence relations for our general discrete system. For the case
with (k2, `2) = (k1, `1) = (k, `) the mapping
(k, `) 7→
(
k̃, ˜̀
)
= (N − `,N − k) (4.1a)
is an involution on the parameters, so we refer to such systems as dual. The self-dual case is
when (k̃, ˜̀) = (k, `), giving k + ` = N . In particular, we consider the case with
k + ` = N, `− k = 1 ⇒ N = 2k + 1, (4.1b)
so we require that N is odd. In this case, we have that Equations (2.5) are invariant under the
change(
φ(i)m,n, α, β
)
7→
(
φ̃(i)m,n, α̃, β̃
)
, where α̃α = 1, β̃β = 1, φ̃(i)m,nφ
(2k−1−i)
m,n = 1. (4.1c)
The self-dual case admits the reduction φ̃
(i)
m,n = φ
(i)
m,n, when α = −β (= 1, without loss of
generality), which we write as
φ(i+k)m,n φ(k−1−i)m,n = 1, i = 0, . . . , k − 1.
The condition
N−1∏
i=0
φ
(i)
m,n = 1 then implies φ
(N−1)
m,n = 1. Therefore the matrices Um,n and Vm,n are
built from k components:
Um,n = diag
(
φ
(0)
m+1,nφ
(k−1)
m,n , . . . , φ
(k−1)
m+1,nφ
(0)
m,n,
1
φ
(k−1)
m+1,n
,
1
φ
(0)
m,nφ
(k−2)
m+1,n
, . . .
. . . ,
1
φ
(k−2)
m,n φ
(0)
m+1,n
,
1
φ
(k−1)
m,n
)
Ωk,
with Vm,n given by the same formula, but with (m + 1, n) replaced by (m,n + 1). In this case
the system (2.6) reduces to
φ
(i)
m+1,n+1φ
(i)
m,n =
1
φ
(k−i−2)
m+1,n φ
(k−i−2)
m,n+1
(
φ
(k−i−2)
m+1,n + φ
(k−i−2)
m,n+1
φ
(k−i−1)
m+1,n + φ
(k−i−1)
m,n+1
)
, for i = 0, 1, . . . , k − 1. (4.2)
Remark 4.1. This reduction has N−1
2 components and is represented by an N ×N Lax pair,
but is not 3D consistent.
Symmetries
Below we give the explicit forms of the self-dual case for N = 3, N = 5 and N = 7. In
each case, we give the lowest order symmetry X1. However, this symmetry does not reduce
to the case of (4.2), but the second symmetry, X2, of the hierarchy generated by the master
symmetries (3.5), is a symmetry of the reduced system.
6 A.P. Fordy and P. Xenitidis
4.1 The case N = 3, with level structure (1, 2; 1, 2)
After the transformation φ
(0)
m,n → 1/φ
(0)
m,n, this system becomes
φ
(0)
m+1,n+1 =
αφ
(1)
m+1,n − βφ
(1)
m,n+1
αφ
(0)
m+1,nφ
(1)
m,n+1 − βφ
(0)
m,n+1φ
(1)
m+1,n
1
φ
(0)
m,n
, (4.3a)
φ
(1)
m+1,n+1 =
αφ
(0)
m,n+1 − βφ
(0)
m+1,n
αφ
(0)
m+1,nφ
(1)
m,n+1 − βφ
(0)
m,n+1φ
(1)
m+1,n
1
φ
(1)
m,n
. (4.3b)
System (4.3) admits two point symmetries generated by{
∂εφ
(0)
m,n = ωn+mφ
(0)
m,n,
∂εφ
(1)
m,n = 0,
{
∂ηφ
(0)
m,n = 0,
∂ηφ
(1)
m,n = ωn+mφ
(1)
m,n,
ω2 + ω + 1 = 0,
and two local generalized symmetries. Here we present the symmetry for the m-direction whereas
the ones in the n-direction follow by changing φ
(i)
m+j,n → φ
(i)
m,n+j
∂t1φ
(0)
m,n = φ(0)m,n
1 + φ
(0)
m+1,nφ
(0)
m,nφ
(0)
m−1,n − 2φ
(1)
m+1,nφ
(1)
m,nφ
(1)
m−1,n
1 + φ
(0)
m+1,nφ
(0)
m,nφ
(0)
m−1,n + φ
(1)
m+1,nφ
(1)
m,nφ
(1)
m−1,n
,
∂t1φ
(1)
m,n = −φ(1)m,n
1− 2φ
(0)
m+1,nφ
(0)
m,nφ
(0)
m−1,n + φ
(1)
m+1,nφ
(1)
m,nφ
(1)
m−1,n
1 + φ
(0)
m+1,nφ
(0)
m,nφ
(0)
m−1,n + φ
(1)
m+1,nφ
(1)
m,nφ
(1)
m−1,n
.
We also have the master symmetry (3.5), which can be written
∂τφ
(0)
m,n = m∂t1φ
(0)
m,n, ∂τφ
(1)
m,n = m∂t1φ
(1)
m,n, ∂τα = α,
which allows us to construct a hierarchy of symmetries of system (4.3) in the m-direction. For
instance, the second symmetry is
∂t2φ
(0)
m,n =
φ
(0)
m,nφ
(1)
m+1,nφ
(1)
m,nφ
(1)
m−1,n
Fm,n
(Sm + 1)
(
(Sm − 1)
(
φ
(0)
m,nφ
(0)
m−1,nφ
(0)
m−2,n
)
Fm,nFm−1,n
)
, (4.4a)
∂t2φ
(1)
m,n =
φ
(1)
m,nφ
(0)
m+1,nφ
(0)
m,nφ
(0)
m−1,n
Fm,n
(Sm + 1)
(
(Sm − 1)
(
φ
(1)
m,nφ
(1)
m−1,nφ
(1)
m−2,n
)
Fm,nFm−1,n
)
, (4.4b)
where
Fm,n := 1 + φ
(0)
m+1,nφ
(0)
m,nφ
(0)
m−1,n + φ
(1)
m+1,nφ
(1)
m,nφ
(1)
m−1,n, (4.4c)
and Sm denotes the shift operator in the m-direction.
The reduced system
The reduced system (4.2) takes the explicit form (first introduced in [5])
φm,nφm+1,n+1(φm+1,n + φm,n+1) = 2, (4.5)
where
φ(0)m,n = φ(1)m,n =
1
φm,n
, β = −α.
Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice 7
With this coordinate, the second symmetry (4.4) takes the form
∂t2φm,n = φm,n
1
P
(1)
m,n
(
1
P
(1)
m+1,n
− 1
P
(1)
m−1,n
)
,
where
P (0)
m,n = φm+1,nφm,nφm−1,n, P (1)
m,n = 2 + P (0)
m,n, (4.6)
first given in [5]. Despite the t2 notation, this is the first of the hierarchy of symmetries of the
reduction (4.5).
4.2 The case N = 5, with level structure (2, 3; 2, 3)
In this case, equations (2.6) take the form
φ
(0)
m+1,n+1 =
φ
(2)
m+1,nφ
(2)
m,n+1
φ
(0)
m,n
αφ
(3)
m+1,n − βφ
(3)
m,n+1
αφ
(2)
m+1,n − βφ
(2)
m,n+1
, (4.7a)
φ
(1)
m+1,n+1 =
1
φ
(1)
m,n
(
αφ
(3)
m+1,n − βφ
(3)
m,n+1
) ( αφ
(3)
m,n+1
φ
(0)
m+1,nφ
(1)
m+1,nφ
(2)
m+1,n
−
βφ
(3)
m+1,n
φ
(0)
m,n+1φ
(1)
m,n+1φ
(2)
m,n+1
)
,
φ
(2)
m+1,n+1 =
αφ
(0)
m+1,n − βφ
(0)
m,n+1
φ
(2)
m,n
(
αφ
(0)
m,n+1φ
(1)
m,n+1φ
(2)
m,n+1φ
(3)
m,n+1 − βφ
(0)
m+1,nφ
(1)
m+1,nφ
(2)
m+1,nφ
(3)
m+1,n
) , (4.7b)
φ
(3)
m+1,n+1 =
φ
(0)
m+1,nφ
(0)
m,n+1
φ
(3)
m,n
αφ
(1)
m+1,n − βφ
(1)
m,n+1
αφ
(0)
m+1,n − βφ
(0)
m,n+1
. (4.7c)
Under the transformation (4.1), the first and last of these interchange, as do the middle pair.
The lowest order generalised symmetry in the m-direction is generated by
∂t1φ
(i)
m,n = φ(i)m,n
(
5A
(i)
m,n
Bm,n
− 1
)
, i = 0, . . . , 3,
where, if we denote F
(i)
m,n = φ
(i)
m−1,nφ
(i)
m,nφ
(i)
m+1,n,
A(0)
m,n = F (0)
m,nF
(1)
m,nF
(2)
m,nφ
(1)
m,nφ
(2)
m,nφ
(3)
m,n, A(1)
m,n = F (0)
m,nF
(1)
m,nF
(2)
m,nF
(3)
m,n
φ
(2)
m,n
φ
(0)
n,m
,
A(2)
m,n = φ(2)m,nφ
(3)
m,n, A(3)
m,n = φ
(0)
m−1,nφ
(0)
m+1,n,
Bm,n =
3∑
j=0
A(j)
m,n + F (0)
m,nF
(1)
m,nφ
(2)
m,n.
The corresponding master symmetry is
∂τφ
(i)
m,n = m∂t1φ
(i)
m,n, i = 0, . . . , 3,
along with ∂τα = 1. This is used to construct a hierarchy of symmetries for the system (4.7). We
omit here the second symmetry as the expressions become cumbersome for the unreduced case.
8 A.P. Fordy and P. Xenitidis
The reduced system
The reduction (4.2) now has components φ
(0)
m,n, φ
(1)
m,n, with φ
(2)
m,n = 1
φ
(1)
m,n
, φ
(3)
m,n = 1
φ
(0)
m,n
, φ
(4)
m,n = 1,
and the 2-component system takes the form
φ
(0)
m+1,n+1φ
(0)
m,n =
1
φ
(0)
m+1,nφ
(0)
m,n+1
(
φ
(0)
m+1,n + φ
(0)
m,n+1
φ
(1)
m+1,n + φ
(1)
m,n+1
)
, (4.8a)
φ
(1)
m+1,n+1φ
(1)
m,n(φ
(0)
m+1,n + φ
(0)
m,n+1) = 2. (4.8b)
Only the even indexed generalised symmetries of the system (4.7) are consistent with this re-
duction. This means that the lowest order generalised symmetry is
∂t2φ
(0)
m,n = φ(0)m,n
P
(0)
m,n
P
(2)
m,n
(
1
P
(2)
m+1,n
− 1
P
(2)
m−1,n
)
, (4.9a)
∂t2φ
(1)
m,n = φ(1)m,n
1
P
(2)
m,n
(
1 + P
(0)
m+1,n
P
(2)
m+1,n
−
1 + P
(0)
m−1,n
P
(2)
m−1,n
)
, (4.9b)
where
P (0)
m,n = φ
(0)
m−1,nφ
(0)
m,nφ
(0)
m+1,nφ
(1)
m,n, (4.10a)
P (1)
m,n = φ
(0)
m−1,n
(
φ(0)m,n
)2
φ
(0)
m+1,nφ
(1)
m−1,nφ
(1)
m,nφ
(1)
m+1,n, (4.10b)
P (2)
m,n = 2 + 2P (0)
m,n + P (1)
m,n. (4.10c)
4.3 The case N = 7, with level structure (3, 4; 3, 4)
The fully discrete system (2.6) and its lower order symmetries (3.4) and master symmetries (3.5)
can be easily adapted to our choices N = 7 and (k, `) = (3, 4). In the same way the correspon-
ding reduced system follows from (4.2) with k = 3. Thus we omit all these systems here and
present only the lowest order symmetry of the reduced system which takes the following form
∂t2φ
(0)
m,n = φ(0)m,n
P
(1)
m,n
P
(3)
m,n
(
1
P
(3)
m+1,n
− 1
P
(3)
m−1,n
)
,
∂t2φ
(1)
m,n = φ(1)m,n
P
(0)
m,n
P
(3)
m,n
(
1 + P
(0)
m+1,n
P
(3)
m+1,n
−
1 + P
(0)
m−1,n
P
(3)
m−1,n
)
, (4.11)
∂t2φ
(2)
m,n = φ(2)m,n
1
P
(3)
m,n
(
1 + P
(0)
m+1,n + P
(1)
m+1,n
P
(3)
m+1,n
−
1 + P
(0)
m−1,n + P
(1)
m−1,n
P
(3)
m−1,n
)
,
where
P (0)
m,n = φ
(0)
m−1,nφ
(0)
m+1,nφ
(1)
m,nφ
(2)
m,n, (4.12a)
P (1)
m,n = φ
(0)
m−1,nφ
(0)
m,nφ
(0)
m+1,nφ
(1)
m−1,n
(
φ(1)m,n
)2
φ
(1)
m+1,nφ
(2)
m,n, (4.12b)
P (2)
m,n = φ
(0)
m−1,n
(
φ(0)m,n
)2
φ
(0)
m+1,nφ
(1)
m−1,n
(
φ(1)m,n
)2
φ
(1)
m+1,nφ
(2)
m−1,nφ
(2)
m,nφ
(2)
m+1,n, (4.12c)
P (3)
m,n = 2 + 2P (0)
m,n + 2P (1)
m,n + P (2)
m,n. (4.12d)
Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice 9
5 Miura transformations and relation to Bogoyavlensky lattices
In this section we discuss Miura transformations for the reduced systems and their symmetries
which bring the latter to polynomial form. In the lowest dimensional case (N = 3) the polyno-
mial system is directly related to the Bogoyavlensky lattice (see (5.2) below), whereas the higher
dimensional ones result in systems which generalise (5.2) to k component systems.
The reduced system in N = 3
The Miura transformation [5]
ψm,n =
P
(0)
m,n
P
(1)
m,n
− 1,
where P
(0)
m,n and P
(1)
m,n are given in (4.6), maps equation (4.5) to
ψm+1,n+1 + 1
ψm,n + ψm,n+1 + 1
+
ψm+1,n
ψm,n+1
= 0, (5.1)
and its symmetry to
∂t2ψm,n = ψm,n(ψm,n + 1)(ψm+2,nψm+1,n − ψm−1,nψm−2,n), (5.2)
which is related to the Bogoyavlensky lattice [1]
∂t2χm,n = χm,n(χm+2,n + χm+1,n − χm−1,n − χm−2,n),
through the Miura transformation
χm,n = ψm+1,nψm,n(ψm−1,n + 1).
The reduced system in N = 5
The Miura transformation
ψ(0)
m,n =
2P
(0)
m,n
P
(2)
m,n
, ψ(1)
m,n =
P
(1)
m,n
P
(2)
m,n
− 1,
where P
(i)
m,n are given in (4.10), maps system (4.8) to
ψ
(0)
m,n+1ψ
(0)
m+1,n+1 + ψ
(0)
m+1,n
(
ψ
(0)
m,n + ψ
(1)
m,n+1
)
ψ
(1)
m,n + 1
=
ψ
(0)
m,n+1ψ
(1)
m+1,n − ψ
(0)
m+1,nψ
(1)
m,n+1
ψ
(0)
m,n+1 + ψ
(1)
m,n+1
,
ψ
(1)
m+1,n+1 + 1
ψ
(1)
m,n + ψ
(1)
m,n+1 + ψ
(0)
m,n+1 + 1
+
ψ
(0)
m+1,n + ψ
(1)
m+1,n
ψ
(0)
m,n+1 + ψ
(1)
m,n+1
= 0, (5.3)
and its symmetry (4.9) to the following system of polynomial equations in which we have sup-
pressed the dependence on the second index n:
∂t2ψ
(0)
m
ψ
(0)
m
=
(
ψ(0)
m + ψ(1)
m
)(
ψ
(0)
m+2ψ
(0)
m+1 − ψ
(0)
m−1ψ
(0)
m−2 + ψ
(0)
m+1 − ψ
(0)
m−1 + ψ
(1)
m+1 − ψ
(1)
m−1
)
−
(
ψ(1)
m + 1
)(
ψ
(1)
m+2ψ
(1)
m+1 − ψ
(1)
m−1ψ
(1)
m−2
)
+
(
ψ(0)
m − 1
)(
ψ
(1)
m+2ψ
(0)
m+1 − ψ
(0)
m−1ψ
(1)
m−2
)
,
10 A.P. Fordy and P. Xenitidis
∂t2ψ
(1)
m
ψ
(1)
m + 1
=
(
ψ(0)
m + ψ(1)
m
)(
ψ
(0)
m+2ψ
(0)
m+1 − ψ
(0)
m−1ψ
(0)
m−2
)
− ψ(1)
m
(
ψ
(1)
m+2ψ
(1)
m+1 − ψ
(1)
m−1ψ
(1)
m−2
)
+ ψ(0)
m
(
ψ
(1)
m+2ψ
(0)
m+1 − ψ
(0)
m−1ψ
(1)
m−2
)
. (5.4)
The above system and its symmetry can be considered as a two-component generalisation of
the equation (5.1) and its symmetry (5.2) in the following sense. If we set, ψ
(0)
m,n = 0 and
ψ
(1)
m,n = ψm,n in (5.3) and (5.4), then they will reduce to equations (5.1) and (5.2), respectively.
The reduced systems for N > 5
It can be easily checked that for each k (N = 2k+ 1), the lowest order symmetry of the reduced
system (4.2) involves certain functions P
(i)
m,n, i = 0, . . . , k, with
P (k)
m,n = 2 + 2
k−2∑
i=0
P (i)
m,n + P (k−1)
m,n ,
which are given in terms of φ
(i)
m,n and their shifts (see relations (4.10) and (4.12)). Then, the
Miura transformation
ψ(i)
m,n =
2P
(i)
m,n
P
(k)
m,n
, i = 0, . . . , k − 2, ψ(k−1)
m,n =
P
(k−1)
m,n
P
(k)
m,n
− 1, (5.5)
brings the symmetries of the reduced system to polynomial form. One could derive the polyno-
mial system corresponding to N = 7 (k = 3) starting with system (4.11), the functions given
in (4.12) and using the corresponding Miura transformation (5.5). The system of differential-
difference equations is omitted here because of its length but it can be easily checked that if we
set ψ
(0)
m,n = 0 and then rename the remaining two variables as ψ
(i)
m,n 7→ ψ
(i−1)
m,n , then we will end
up with system (5.4).
This indicates that every k component system is a generalisation of all the lower order ones,
and thus of the Bogoyavlensky lattice (5.2). To be more precise, if we consider the caseN = 2k+1
along with the k-component system, set variable ψ
(0)
m,n = 0 and then rename the remaining
ones as ψ
(i)
m,n 7→ ψ
(i−1)
m,n , then the resulting (k − 1)-component system is the reduced system
corresponding to N = 2k− 1. Recursively, this means that it also reduces to the N = 3 system,
i.e., equation (5.2). Other systems with similar behaviour have been presented in [4].
Acknowledgements
PX acknowledges support from the EPSRC grant Structure of partial difference equations with
continuous symmetries and conservation laws, EP/I038675/1.
References
[1] Bogoyavlensky O.I., Integrable discretizations of the KdV equation, Phys. Lett. A 134 (1988), 34–38.
[2] Fordy A.P., Xenitidis P., ZN graded discrete Lax pairs and discrete integrable systems, arXiv:1411.6059.
[3] Fordy A.P., Xenitidis P., ZN graded discrete Lax pairs and integrable difference equations, J. Phys. A:
Math. Theor. 50 (2017), 165205, 30 pages.
[4] Mar̀ı Beffa G., Wang J.P., Hamiltonian evolutions of twisted polygons in RPn, Nonlinearity 26 (2013),
2515–2551, arXiv:1207.6524.
[5] Mikhailov A.V., Xenitidis P., Second order integrability conditions for difference equations: an integrable
equation, Lett. Math. Phys. 104 (2014), 431–450, arXiv:1305.4347.
[6] Yamilov R., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen.
39 (2006), R541–R623.
https://doi.org/10.1016/0375-9601(88)90542-7
https://arxiv.org/abs/1411.6059
https://doi.org/10.1088/1751-8121/aa639a
https://doi.org/10.1088/1751-8121/aa639a
https://doi.org/10.1088/0951-7715/26/9/2515
https://arxiv.org/abs/1207.6524
https://doi.org/10.1007/s11005-013-0668-8
https://arxiv.org/abs/1305.4347
https://doi.org/10.1088/0305-4470/39/45/R01
1 Introduction
2 ZN-graded Lax pairs
2.1 Quotient potentials
3 Differential-difference equations as symmetries
3.1 Symmetries in potentials variables
4 The self-dual case
4.1 The case N=3, with level structure (1,2;1,2)
4.2 The case N=5, with level structure (2,3;2,3)
4.3 The case N=7, with level structure (3,4;3,4)
5 Miura transformations and relation to Bogoyavlensky lattices
References
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