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A noncommutative discrete potential KdV lift

TL;DR: In this article, a Grassmann extension of the Yang-Baxter map is presented, which can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation.
Abstract: In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in the work of Kouloukas and Papageorgiou [J. Phys. A: Math. Theor. 42, 404012 (2009)] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix. Moreover, we show that it can be squeezed down to a novel system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogs of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability.In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in the work of Kouloukas and Papageorgiou [J. Phys. A: Math. Theor. 42, 404012 (2009)] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix. Moreover, we show that it can be squeezed down to a novel system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogs of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability.

Summary (2 min read)

1 Introduction

  • Noncommutative extensions of integrable systems have been of extensive interest over the last few decades.
  • The main approach is based on the symmetry analysis of the corresponding equations [25, 27].
  • In section 4, the authors show that the constructed Yang-Baxter map can be reduced to a discrete quad-graph system, which constitutes a noncommutative extension of the dpKdV equation.

2.1 Grassmann algebra

  • Here, the authors give all the definitions related to Grassmann algebras that are essential for this paper.
  • For more information on Grassmann analysis one can consult [3].
  • The blocks P and L are matrices with even entries, while Π and Λ possess only odd entries.
  • Henceforth, the authors adopt the following notation:.

2.3 Integrable equations on quad-graphs

  • The three dimensional consistency property is a sufficient (but not necessary) condition for a quad-graph equation to possess a Lax representation [4, 23].
  • That is, the Lax matrix of the initial equation is also a Lax matrix of its lift.
  • As in the case of Yang-Baxter maps, the authors can consider quadrilateral equations and Lax representations defined on Grassmann variables.
  • 2Similarly, the invariants of the transfer maps [33] are derived from the supertrace of the corresponding monodromy matrix.

3 Noncommutative dpKdV lift

  • In general, the Lax matrices of Yang-Baxter maps and quadrilateral equations are not unique. , x := (x1, x2), is also a Lax matrix for both the Yang-Baxter map (7) and the dpKdV equation.
  • The authors have to notice that this is not the only possible extension of L̃a satisfying these two properties.

4 Squeeze down to a noncommutative dpKdV system

  • This system is defined with three fields on each vertex of an elementary quadrilateral, one even and two odd.
  • Similarly to [19, 26], the authors prove the following proposition for a specific class of Yang-Baxter maps.
  • Finally, substituting these values to (17), the authors obtain (21).

5 Integrable commutative analogues

  • If the authors consider the version of the map before simplifying the entries ξ1 and η2 in (16c) and (16h), respectively, using the properties of odd variables (see (14)), then the corresponding map with all its variables being considered as even satisfies the Yang-Baxter equation.
  • Given a Yang-Baxter map, the authors are mostly interested in studying the integrability of the corresponding transfer maps which occur by considering periodic initial value problems on quadrilateral lattices [17, 24, 33, 34].
  • Furthermore, the Sklyanin bracket ensures the involutivity of the integrals derived by the trace of the corresponding monodromy matrix, which consists of products of local Lax matrices.
  • That is, integrals I1, I2 and I4, while I3 was derived simply by inspection.

6 Conclusions

  • The authors constructed a Grassmann extension of a Yang-Baxter map, namely map (15)-(16), from a “lift” of the famous dpKdV equation, and they proved that it possesses the Yang-Baxter property.
  • The authors showed that this map can be squeezed down to a Grassmann quad-graph dpKdV system, i.e. system (20), with three fields on each vertex of the quad-graph; one even and two odd.
  • Finally, the authors studied the commutative analogues of these constructions, and they proved that they satisfy similar properties.
  • The authors believe that their results can be extended in several ways.
  • In [10, 15], the associated Lax matrices of the derived Yang-Baxter maps constitute Darboux transformations of certain PDEs of NLS and generalised KdV type, respectively.

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Citation for published version
Konstantinou-Rizos, Sotiris and Kouloukas, Theodoros (2018) A noncommutative discrete potential
KdV lift. Journal of Mathematical Physics, 59 (6). ISSN 0022-2488.
DOI
https://doi.org/10.1063/1.5041947
Link to record in KAR
https://kar.kent.ac.uk/67389/
Document Version
Pre-print

arXiv:1611.08923v1 [nlin.SI] 27 Nov 2016
A noncommutative discrete potential KdV lift
S. Konstantinou-Rizos
1,2
and T. E. Kouloukas
3
1
Institute of Mathematical Physics and Seismodynamics, Chechen State University, Russia
2
Faculty of Mathematics and Computer Technology, Chechen State University, Russia
3
School of Mathematics, Statistics & Actuarial Science, University of Kent, UK
November 29, 2016
Abstract
In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared
in [16] and can be conside red as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation.
This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix.
Moreover, we show that it can be squeezed down to a system of lattice equations which posse sses a
Lax representation and whose bosonic limit is the dpKdV equa tion. Finally, we consider commutative
analogues of the constructed Yang-Baxter map and its associated quad-graph system, and we discus s
their integrability.
PACS numbers: 02.30.Ik
Mathematics Subject Classification: 15A75, 35Q53, 39A14, 81R12.
Keywords: Noncommutative dpKdV, Grassmann extensions of Yang-Baxter maps,
Grassmann algebras, Grassmann extensions of discrete integrable systems.
1 Introduction
Noncommutative extensions of integrable systems have been of extensive interest over the last few decades.
Recently, in [11], a method was presented for constructing super (Grasmmann-extended) differential-
difference and difference-difference systems via noncommutative extensions of Darboux transformations.
This motivated f urther study of discrete integrable systems on Grassmann algebras [35, 36, 37], as well
as Grassm an n extensions of Yang-Baxter maps [10, 15].
The theory of Yang-Baxter maps, namely set-theoretical solutions of the Yang-Baxter equation, has a
fundamental role in the theory of integrable systems. The study of such solutions was formally proposed
by Drinfel’d in [8]. Yang-Baxter maps and several connections with discrete integrable systems were
extensively s tudied over the past couple of decades (indicatively we refer to [2, 6, 9, 13, 27, 29, 33,
34]). Lax representations [32] of Yang-Baxter maps are of particular interest, since they are associated
with refactorization problems of polynomial matrices, invariant spectral curves of transfer maps [33,
34], r-matrix Poisson structures [16, 18], and they also possess a natural connection with integrable
skonstantin84@gmail.com, skonstantin@chesu.ru
T.E.Kouloukas@kent.ac.uk
1

partial d ifferential equations through Darboux transformations [14, 20]. Recently, a “lifting” method
for constructing two-field equations, which can be viewed as Yang-Baxter maps, via equations on quad-
graphs, was presented in [26].
The relation between the Yang-Baxter property of maps and the multi-dimensional consistency prop-
erty of integrable equations on quad-graphs is well established. The main approach is based on the
symmetry analysis of the corresponding equations [25, 27]. From another point of view, under some
conditions 3D consistent equations can be lifted to fou r dimensional Yang-Baxter maps. This lifting pro-
cedure has been described in [26]. Conversely, Yang-Baxter maps of a specific form can be squeezed down
to lattice equations and systems that admit a Lax representation derived from the Lax matrix of the
corresponding Yang-Baxter map. Motivated by these results and working in the direction of extending
the theory of Yang-Baxter maps to the noncommutative case, in this paper, we constr uct a Grassmann
extension of a Yang-Baxter map, first derived in [16], and we show that it possesses the Yang-Baxter
property. Furthermore, we us e this map to construct a noncommutative extension of the well-celebrated
dpKdV equation [12, 22, 24], also referred to as H
1
equation in the ABS list [1]. Both the Yang-Baxter
map and its associated quad-graph system, which we construct, admit Lax r epresentations with 3 × 3
Lax m atrices and th eir bosonic limit leads to the original discrete systems.
Finally, we consider all the variables of the constructed 3 × 3 Lax matrices to be commutative, and
we derive a higher-dimensional commu tative extension of the original Yang-Baxter map. This extension
constitutes a symplectic map with respect to the Sklyanin bracket, and completely integrable in the
Liouville sense. As in the noncommutative case, this map can be squeezed down to a corresponding
integrable quad-graph system.
The paper is organised as follows. In the next s ection, we begin with some preliminaries on Grassmann
algebras, Yang-Baxter maps, quadrilateral equations and Lax representations. In section 3, we present
the construction of the Grassmann extension of the lift of the dpKdV equation, and we show that it
satisfies the Yang-Baxter equation. I n section 4, we show that the constructed Yang-Baxter map can be
reduced to a discrete quad-graph system, which constitutes a n on commutative extension of the dpKdV
equation. The Lax representation of th is system is derived fr om the corresponding Lax matrix of the
Yang-Baxter map. Section 5 deals with commutative analogues of the systems presented in sections 3
and 4. Finally, in section 6, we close with some remarks and perspectives for future work.
2 Preliminaries
2.1 Grassmann algebra
Here, we give all the definitions related to Grassmann algebras that are essential for this paper. However,
for more information on Grassmann analysis one can consult [3].
Consider G to be a Z
2
-graded algebra over C. Thus, G, as a lin ear space, is a direct sum G = G
0
G
1
(mod 2), such that G
i
G
j
G
i+j
. The elements of G that belong either to G
0
or to G
1
are called
homogeneous, th e ones in G
1
are called odd (or fermionic), w hile those in G
0
are called even (or bosonic).
The parity |a| of an even homogeneous element a is 0, while it is 1 for o dd homogeneous elements,
by definition. The parity of the product |ab| of two homogeneous elements is a sum of their parities:
|ab| = |a| + |b|. Now, for any homogeneous elements a and b, Grassmann commutativity means that
ba = ( 1)
|a||b|
ab . Th is implies that if α G
1
, th en α
2
= 0, and αa = , for any a G
0
.
The notions of the determinant and the trace of a m atrix in G are defined for square matrices, M, of
2

the following block-form
1
M =
P Π
Λ L
.
The blocks P and L are matrices with even entries, while Π and Λ possess only odd entries. In particular,
the superdeterminant of M , which is usually denoted by sdet(M), is defined to be the following quantity
sdet(M) = det(P ΠL
1
Λ) det(L
1
) = det(P
1
) det(L ΛP
1
Π),
where det(·) is the usu al determinant of a matrix, w hile the supertrace, which is us ually denoted by
str(M), is defined as
str(M) = tr(P ) tr(L),
where tr(·) is the usual trace of a matrix.
Henceforth, we adopt the following notation: We denote all even variables in G
0
by Latin letters,
whereas for odd variables in G
1
we use Greek letters.
2.2 Yang-Baxter equation and Lax representations in the Grassmann case
Let S : V
G
× V
G
V
G
× V
G
be a map
((x, χ), (y, ψ))
S
7→ ((u, ξ), (v, η)) , (1)
where V
G
= {(a, α) | a G
0
, α G
1
}. Map (1) is said to be a Grassmann extended Yang-Baxter map,
if it satisfies the Yang-Baxter equation:
S
12
S
13
S
23
= S
23
S
13
S
12
,
where the maps S
ij
: V
G
× V
G
× V
G
V
G
× V
G
× V
G
, i, j = 1, 2, 3, i 6= j, are defined by the following
relations
S
12
= S × id, S
23
= id × S and S
13
= π
12
S
23
π
12
,
where π
12
is the involution defined by π
12
((x, χ), (y, ψ), (z, ζ)) = ((y, ψ), (x, χ), (z, ζ)). It is obvious from
the above definition that definitions of Yang-Baxter maps in the Grassmann and commutative cases
coincide. The only difference is the set of the objects of the maps. Additionally, map (1) is called
reversible if S
21
S
12
= id; map S
21
is defined as S
21
= π
12
S
12
π
12
.
Furthermore, if two parameters a, b G
0
are involved in the definition of (1), n amely we have a map
S
a,b
: ((x, χ), (y, ψ)) 7→ ((u, ξ), (v, η)) , (2)
satisfying the parametric Yang-Baxter equation
S
12
a,b
S
13
a,c
S
23
b,c
= S
23
b,c
S
13
a,c
S
12
a,b
, (3)
we shall be using the term Grassmann extended parametric Yang-Baxter map.
According to [32], a Lax matrix of the parametric Yang-Baxter map (2), is a matrix L, with Grassmann-
valued entries in this case, depending on the point (x, χ) V
G
, a parameter a and a spectral parameter
λ, such that
L
a
(u, ξ; λ)L
b
(v, η; λ) = L
b
(y, ψ; λ)L
a
(x, χ; λ). (4)
1
Note that the block matrices are not necessarily square matrices.
3

The refactorization equation (4) d oes not always admit a unique solution with respect to u, ξ, v, η. In
fact, if (4) is equivalent to ((u, ξ), (v, η)) = S
a,b
((x, χ), (y, ψ)), then the Lax matrix L is said to be strong
[18]. Moreover , general s olutions of (4) are not always Yang-Baxter maps. For the Yang-Baxter property
one may use the following trifactorisation criterion: If the following matrix refactorisation problem
L
a
(u, ξ;λ)L
b
(v, η; λ)L
c
(w, γ; λ) = L
a
(x, χ; λ)L
b
(y, ψ; λ)L
c
(z, ζ; λ),
implies
(u, ξ) = (x, χ), (v, η) = (y, ψ) and (w, γ) = (z, ζ),
then map (2) is a Yang-Baxter map [16, 33].
Similarly to the commutative case, the quantity str(L
b
(y, ψ; λ)L
a
(x, χ; λ)) constitutes a generating
function of invariants for map S
a,b
2
. This can be verified by applying the supertrace to both parts of the
Lax equ ation (4).
2.3 Integrable equations on quad-graphs
Let w be a function of two discrete variables n and m. Let also S and T be the shift operators in the
n and m direction of a two-dimensional lattice, respectively. We shall be using the notation: w
00
w,
w
ij
= S
i
T
j
f; for example, w
10
= w(n + 1, m), w
01
= w(n, m + 1) and w
11
= w(n + 1, m + 1).
Next, we consider equations defined at the vertices of an elementary quadrilateral (see Figure 1),
Q(w, w
10
, w
01
, w
11
; a, b) = 0. (5)
w
w
10
w
01
w
11
a
a
b b
n
m
Figure 1: Quad-Graph.
A Lax representation of (5) is an equation
L(w
01
, w
11
, a)M (w, w
01
, b) = M (w
10
, w
11
, b)L(w, w
10
, a), (6)
for a pair of matrices L, M, equivalent to (5). In many cases, as in the one we f ocus in this paper, L = M.
The three dimensional consistency property is a sufficient (but not necessary) condition for a quad-graph
equation to possess a Lax representation [4, 23]. In the case of a lift of a quad-graph equation (or system
of equations) to a Yang-Baxter map, as presented in [26], it can be proven that the Lax matrix of the
Yang-Baxter map coincides with the Lax matrix (for L = M in (6)) of the corresponding equation [19].
In many cases, as in the case we investigate in this paper, the converse also holds. That is, the Lax
matrix of the initial equation is also a Lax matrix of its lift. As in the case of Yang-Baxter maps, we can
consider quad rilateral equ ations and Lax representations defined on Grassmann variables.
2
Similarly, the invariants of the transfer maps [33] are derived from the supertrace of the corresponding monodromy
matrix.
4

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Q1. What are the contributions in this paper?

In this paper, the authors construct a Grassmann extension of a Yang-Baxter map which first appeared in [ 16 ] and can be considered as a lift of the discrete potential Korteweg-de Vries ( dpKdV ) equation. Moreover, the authors show that it can be squeezed down to a system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, the authors consider commutative analogues of the constructed Yang-Baxter map and its associated quad-graph system, and they discuss their integrability.