Families of Integrable Equations
We present a method to obtain families of lattice equations. Specifically we focus on two of such families, which include 3-parameters and their members are connected through Bäcklund transformations. At least one of the members of each family is integrable, hence the whole family inherits some inte...
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| Цитувати: | Families of Integrable Equations / P. Kassotakis, M. Nieszporski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 33 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859717236472676352 |
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| author | Kassotakis, P. Nieszporski, M. |
| author_facet | Kassotakis, P. Nieszporski, M. |
| citation_txt | Families of Integrable Equations / P. Kassotakis, M. Nieszporski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 33 назв. — англ. |
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| description | We present a method to obtain families of lattice equations. Specifically we focus on two of such families, which include 3-parameters and their members are connected through Bäcklund transformations. At least one of the members of each family is integrable, hence the whole family inherits some integrability properties.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 100, 14 pages
Families of Integrable Equations?
Pavlos KASSOTAKIS † and Maciej NIESZPORSKI ‡
† Department of Mathematics and Statistics University of Cyprus,
P.O. Box: 20537, 1678 Nicosia, Cyprus
E-mail: kassotakis.pavlos@ucy.ac.cy, pavlos1978@gmail.com
‡ Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski,
ul. Hoża 74, 00-682 Warszawa, Poland
E-mail: maciejun@fuw.edu.pl
Received May 23, 2011, in final form October 20, 2011; Published online October 28, 2011
http://dx.doi.org/10.3842/SIGMA.2011.100
Abstract. We present a method to obtain families of lattice equations. Specifically we
focus on two of such families, which include 3-parameters and their members are connected
through Bäcklund transformations. At least one of the members of each family is integrable,
hence the whole family inherits some integrability properties.
Key words: integrable lattice equations; Yang–Baxter maps; consistency around the cube
2010 Mathematics Subject Classification: 82B20; 37K35; 39A05
1 Introduction
Discrete mathematics returned on the interest of mathematicians at the beginning of the 20th
century. Poincaré, Birkhoff, Ritt (1924) [1], Julia, Fatou (1918–1923) [2, 3] and many others
saw the necessity of exploring the discrete scene. Unfortunately, this trend was paused through
the two big wars and only after 1960, keeping pace with the revolution caused by the discovery
of soliton from Zabusky and Kruskal [4], mathematicians started to investigate discrete systems
in the context of integrable systems.
It was the work of Hirota [5], as well as Ablowitz et al. [6] and separately Capel and his
school [7], which introduced lattice and differential difference analogues of many integrable
PDE’s. The introduction of discrete versions of integrable ODE’s, surprisingly, came later
with the QRT family of mappings by Quispel, Roberts and Thomson [8] and by the work of
Papageorgiou et al. [9, 10], where Liouville integrable maps [11] were obtained by imposing
periodic staircase initial data on integrable lattices. Another way to obtain integrable mappings
from an integrable lattice equation was suggested in series of papers [12, 13, 14]. Actually
with this procedure one can get involutive mappings (composition of the map with itself is the
identity map) which are set theoretical solutions of the quantum Yang–Baxter equation the so
called Yang–Baxter maps [15, 16, 12, 17]. As in our previous work [18], we focus here on the
inverse procedure, i.e. how to obtain integrable lattice equations from involutive mappings that
may or may not satisfy the Yang–Baxter equation.
The main result of the paper is that the procedure we have in mind can lead to families of
equations. It is necessary to mention that points of the lattice may not be related in a unique
way. The members of families are related by a Bäcklund transformation (see Section 6) and
since in considered cases at least one of the members is integrable, the whole family inherits
?This paper is a contribution to the Proceedings of the Conference “Symmetries and Integrability of
Difference Equations (SIDE-9)” (June 14–18, 2010, Varna, Bulgaria). The full collection is available at
http://www.emis.de/journals/SIGMA/SIDE-9.html
mailto:kassotakis.pavlos@ucy.ac.cy
mailto:pavlos1978@gmail.com
mailto:maciejun@fuw.edu.pl
http://dx.doi.org/10.3842/SIGMA.2011.100
http://www.emis.de/journals/SIGMA/SIDE-9.html
2 P. Kassotakis and M. Nieszporski
some properties from the distinguished member. We stress that solutions of each member of the
family can be obtained from solutions of the integrable member by discrete quadratures (which
can be regarded as sort of Bäcklund transformation) and in this sense each member of the family
is integrable. However, we discuss here hallmarks of integrability of the members of the family
such as consistency around the cube property or τ -function formulation. Notion of the family
of discrete integrable systems should not be confused with notion of hierarchies of integrable
systems. The later notion was widely investigated in the literature whereas for the former one
we can indicate only the articles that investigate the family of discrete KdV equations [19] and
the family of discrete Boussinesq equations [20, 21, 22].
We discuss here two examples, the first one is continuation of our previous paper [18]. We
introduce a family of difference equations associated with type III of maps discussed in [12, 13]
(we introduced families related to types IV and V in [18]). Example of the map of type III is
a map C2 3 (u, v) 7→ (U, V ) ∈ C2
U = v
pu− qv
qu− pv
, V = u
pu− qv
qu− pv
.
and the three parameter family of equations (see Section 3) reads
ψ12 = ψ + a ln
pu− qv
qu− pv
+
(
p2 − q2
) [
b
uv
qu− pv
− c 1
pu− qv
]
, (1.1)
where u and v are given implicitly by
a lnu+ p
(
bu+ c
1
u
)
= ψ1 + ψ, a ln v + q
(
bv + c
1
v
)
= ψ2 + ψ, (1.2)
function ψ is dependent variable on Z2 and we denote ψ(m,n) =: ψ, ψ(m+1, n) =: ψ1, ψ(m,n+
1) =: ψ2, ψ(m+ 1, n+ 1) =: ψ12, p := p(m) and q := q(n) are given functions of a single variable
and a, b and c are arbitrary constants (we assume that one of the constants a, b or c is not equal
to zero). However, ought to possible branching in formulas (1.2), the system (1.1), (1.2) needs
specifying (as it was pointed us by Professors Frank Nijhoff and Yuri Suris). The specification
is achieved by demanding that functions u and v obey
u2 = v
pu− qv
qu− pv
, v1 = u
pu− qv
qu− pv
. (1.3)
After this specification there is still some freedom left in finding solutions of (1.1), (1.2) for
given initial conditions on ψ. The freedom lies in finding the initial conditions for u and v out
of initial conditions on ψ by means of (1.2). The solution need not to be unique.
All the equations within the family are consistent around the cube (for the consistency around
the cube property see [23, 24, 25, 26], notice we resign from multiaffinity assumption of pa-
per [25]). We find especially interesting the fact that we obtain examples of lattice equations
together with transformations which can be regarded as Bäcklund transformations but not in
the usual sense; we usually require Bäcklund transformation to be linearisable (see Definition 5)
and this requirement is violated in these examples. Therefore Lax pair could not be easily found
from this sort of Bäcklund transformation and it is not clear if the Lax pair exists in these cases.
Members of the family are Hirota’s sine-Gordon equation (choice of parameters b = 0 = c)
referred also to as lattice potential modified KdV [27, 28, 29, 30, 31, 14, 32] (see Section 2 where
we discuss various forms of lattice equations)
p(xx1 + x2x12) = q(xx2 + x1x12)
Families of Integrable Equations 3
and lattice Schwarzian KdV [28] in a disguise, see Section 2 (choice of parameters a = 0 = b or
a = 0 = c)
p2(y12 + y1)(y2 + y) = q2(y12 + y2)(y1 + y).
In the second example we go away from the maps of papers [12, 13] and consider the map
U = v + k
(
1− v
u
)
, V = u+ k
(
−1 +
u
v
)
,
which gives also a 3 parameter family of equations (see Section 5) including Hirota’s KdV lattice
equation [5]
x12 − x = κ
(
1
x2
− 1
x1
)
and two further bilinear equations
y1y − y12y1 = κ(y12y + y1y2), z12z + z1z2 = z12z2 + z12z1.
In this case an interesting fact is that the procedure yields τ -function representation of the
family (see e.g. [19])
τ112τ − κτ11τ2 = τ12τ1, τ122τ + κτ22τ1 = τ12τ2.
In Section 2, we give an overview of point transformations, Bäcklund transformations and
difference substitutions and touch the issue of equivalence of lattice equations. We proceed
in Section 3 where we present the method that leads to families of lattice equations. In Sec-
tion 4 we relate our findings to some results of the papers [12, 13], followed by Section 5 where we
deal with Hirota’s KdV lattice equation. Then we explain how to get Bäcklund transformation
between members of the families (Section 6) and we end the paper with some conclusions and
perspectives for future work.
2 Point transformations, difference substitutions, Bäcklund
transformations and equivalence of lattice equations
Before we start we would like to give some definitions and recall some well known relations [29,
30, 31, 19, 32] between equations that appear in the article (terminology used by various au-
thors is far from being unified). Let us consider k dependent variables of n independent ones:
ui(m1, . . . ,mn), i = 1, . . . , k. We denote M ≡ (m1, . . . ,mn).
Definition 1 (change of independent variables). By change of independent variables we under-
stand the bijection f : Zn → Zn
m̃i = f i(M), i = 1, . . . , n.
2D examples are m̃1 = m1, m̃2 = m1 +m2, or m̃1 = m1 + 2m2, m̃2 = m1 +m2.
Definition 2 (point transformations not altering independent variables). By point transfor-
mation not altering independent variables we understand an invertible map F between subsets
of Ck
ũi(M) = F i
(
u1(M), . . . , uk(M);M
)
, i = 1, . . . , k.
4 P. Kassotakis and M. Nieszporski
Definition 3 (equivalence of lattice equations). Two lattice equations are equivalent if and only
if there exists composition of point transformation with change of independent variables which
maps solutions of one equation to solutions of the second one.
Examples of various disguises of the same equation are
• Hirota’s sine-Gordon equation
q sin(ψ12 + ψ − ψ1 − ψ2) = p sin(ψ12 + ψ + ψ1 + ψ2)
turns into
(H30) : p(xx1 + x2x12) = q(xx2 + x1x12) (2.1)
H30 equation from ABS list [25] by means of point transformation x = im+ne2i(−1)
nψ.
H30 in turn can be transformed into lattice potential modified KdV
p(ww1 − w2w12) = q(ww2 − w1w12)
by substitution x = im+nw.
• Schwarzian KdV equation (or cross ratio equation, or equation Q10 on ABS list)
(z12 − z1)(z2 − z)
(z12 − z2)(z1 − z)
=
q2
p2
under the point transformation z = (−1)m+ny turns into
(A10) : p2(y12 + y1)(y2 + y) = q2(y12 + y2)(y1 + y) (2.2)
which in the paper [25] got its own name A10.
Definition 4 (difference substitutions). Let j points M i, i = 1, . . . , j of a lattice are given. By
difference substitution of order j we understand a transformation
ũi(M) = F i
(
u1
(
M1
)
, . . . , uk
(
M1
)
, . . . , u1
(
M j
)
, . . . , uk
(
M j
)
;M
)
, i = 1, . . . , k.
Every point transformation is difference substitutions of order 1. Standard examples of
difference substitution (of order 2, 3 and 4 respectively) are
• potential relation
v =
1
α− β
(u2 − u1)
between lattice potential KdV
(u12 − u)(u1 − u2) = α2 − β2
and Hirota’s difference KdV
v12 − v =
α+ β
α− β
(
1
v1
− 1
v2
)
;
Families of Integrable Equations 5
• Miura-type transformation
v =
βψ2 − αψ1
(β − α)ψ
between H30 (Hirota’s sine-Gordon or lattice modified potential KdV)
α(ψ2ψ12 − ψψ1) = β(ψ1ψ12 − ψψ2)
and Hirota’s difference KdV;
• and finally the introduction of τ function
v =
τ12τ
τ1τ2
,
which transform every solution of the compatible system
τ112τ − κτ11τ2 = τ12τ1, τ122τ + κτ22τ1 = τ12τ2
to solution of Hirota’s difference KdV.
To the end we propose draft definition of Bäcklund transformation which is convenient for our
purposes. However we are aware that the definition is not exhaustive (some transformation that
deserve this name can be not covered by the definition).
Definition 5 (Bäcklund transformations (in narrow sense)). By Bäcklund transformation
we understand here a transformation between two equations F (u12, u1, u2, u) = 0 and
F̃ (ũ12, ũ1, ũ2, ũ) = 0
ũ1 = f(ũ, u, u1), ũ2 = g(ũ, u, u2),
which is invertible to
u1 = f̃(u, ũ, ũ1), u2 = g̃(u, ũ, ũ2),
where functions f and g are fractional linear in ũ and functions f̃ , g̃ are function fractional
linear in u.
A classical example of Bäcklund transformation between
p(xx1 + x2x12)− q(xx2 + x1x12) = 0
and
p2(y12 + y1)(y2 + y) = q2(y12 + y2)(y1 + y)
is the transformation
y1 + y = px1x, y2 + y = qx2x. (2.3)
3 Outline of the method
We consider the Z2 lattice together with its horizontal edges (which can be viewed as set of
ordered pair of points of Z2, i.e. Eh =
{
((m,n), (m+ 1, n))|(m,n) ∈ Z2
}
) and the vertical ones
(Ev =
{
((m,n), (m,n+ 1))|(m,n) ∈ Z2
}
). We take into account a function u which is given
on horizontal edges u : Eh → C and a function v given on vertical ones v : Ev → C. Shift
operators T1 and T2 act on horizontal edges in standard way T1((m,n), (m + 1, n)) := ((m +
1, n), (m+ 2, n)), T2((m,n), (m+ 1, n)) := ((m,n+ 1), (m+ 1, n+ 1)) (and similarly for vertical
edges). We use convention to denote shift action on a function by subscripts T1u := u1.
Now, the outline of the method we developed in [18] can be presented as follows.
6 P. Kassotakis and M. Nieszporski
3.1 From equations to involutive maps. Idea system
Take a function x given on vertices of the lattice and which obeys H30 equation
p(xx1 + x2x12) = q(xx2 + x1x12). (3.1)
Introduce fields u and v given on horizontal and vertical edges respectively (fields u and v are
actually the invariants of a symmetry group of the lattice equation (3.1) as it was shown in [13])
u = xx1, v = xx2.
We get
u2u = v1v, p(u2 + u) = q(v1 + v)
and we arrive at the system of equations
u2 = v
pu− qv
qu− pv
, v1 = u
pu− qv
qu− pv
. (3.2)
The main idea is to investigate system (3.2) rather than equation (3.1) itself. We dare to refer to
the system (3.2) as to 2D Idea system III. The point is that the system (3.2) admits, as we shall
see, three parameter family of potentials ψ given on vertices of the lattice. Every “potential
image” of (3.2) we refer to as idolon (adopting Plato terminology of Ideas and idolons).
First we apply the standard procedure for reinterpretation of equations on a lattice as a map.
The reinterpretation is based on identification (see Fig. 1)
u(m,n) = u, v(m,n) = v, u(m,n+ 1) = U, v(m+ 1, n) = V, (3.3)
which turns system (3.2) into C2 → C2 map
U = v
pu− qv
qu− pv
, V = u
pu− qv
qu− pv
. (3.4)
We arrive at an involutive Yang–Baxter map that belongs to family of maps denoted by FIII
(see [12]).
Figure 1. Variables on edges of a Z2 lattice (left picture) and arguments and values of a C2 7→ C2 map
(right picture).
3.2 Finding functions such that F (U) + G(V ) = f(u) + g(v)
The next step is to find such functions F and G such that for the map (3.4)
F (U) +G(V ) = f(u) + g(v). (3.5)
Families of Integrable Equations 7
holds. Anticipating facts, the functions will allow us to introduce a family of potentials in the
next subsection. Differentiation of (3.5) with respect to u and v yields
−F ′′(U)qU2
(
pU2 − 2qUV + pV 2
)
+ F ′(U)2(qU − pV )qUV
+G′′(V )pV 2
(
qU2 − 2pUV + qV 2
)
+G′(V )2(qU − pV )pUV = 0.
The equation above should hold for every value of U and V respectively. The equation has the
form
−pqU4F ′′(U) + 2q2U2(UF ′′(U) + F ′(U))V − pqU(UF ′′(U) + F ′(U))V 2
+ α(V )U2 + β(V )U + γ(V ) = 0,
so F (U) must satisfy (necessary but not sufficient condition) the ODE
−pqU4F ′′(U) + c2U
2 + c1U + c0 = 0.
with some constants c1, c2 and c0. Similarly we get
pqV 4G′′(V ) + d2V
2 + d1V + d0 = 0.
Checking obtained by this way solutions we obtain
F (U) +G(V ) = a ln(U/V ) + b(pU − qV ) + c
( p
U
− q
V
)
+ d
and we find that for the map (3.4) the following equality holds
a ln(U/V ) + b(pU − qV ) + c
( p
U
− q
V
)
= −
[
a ln(u/v) + b(pu− qv) + c
(p
u
− q
v
)]
. (3.6)
3.3 Potentials of the Idea systems. Idolons
Returning to equations on the lattice (by means of (3.3)) one can rewrite (3.6) as
(T2 + 1)
(
a lnu+ bpu+ c
p
u
+ d
)
= (T1 + 1)
(
a ln v + bqv + c
q
v
+ d
)
.
It means there exists function ψ such that
a lnu+ p
(
bu+ c
1
u
)
+ d = ψ1 + ψ, a ln v + q
(
bv + c
1
v
)
+ d = ψ2 + ψ (3.7)
where a, b, c and d are arbitrary constants (we assume that one of the constants a, b, c is not
equal zero). The constant d can be always removed by redefinition ψ → ψ + 1
2d and we neglect
it
a lnu+ p
(
bu+ c
1
u
)
= ψ1 + ψ, a ln v + q
(
bv + c
1
v
)
= ψ2 + ψ. (3.8)
System (3.7) and Idea system (3.2) give rise to
ψ12 = ψ + a ln
pu− qv
qu− pv
+
(
p2 − q2
) [
b
uv
qu− pv
− c 1
pu− qv
]
, (3.9)
so we get three parameter family of equations. Note that in general, (3.2) does not follows
from (3.7) and (3.9) and therefore we will treat (3.2) as an additional condition that must be
satisfied. As we have said in the introduction, choice of parameters b = 0 = c leads to equation
8 P. Kassotakis and M. Nieszporski
H30 (2.1) whereas choice of parameters either a = 0 = b or a = 0 = c leads to equation A10 (2.2).
Every such potential representation of the Idea system we refer to as idolon of the Idea system.
To the end let us write another idolon. Namely, a = 0 yields the equation
ψ2 − ψ1
ψ12 − ψ
=
p2 + q2
p2 − q2
− pq
p2 − q2
(u
v
+
v
u
)
,
where u and v are solutions of the following quadratic equations
p
(
bu2 + c
)
= (ψ1 + ψ)u, q
(
bv2 + c
)
= (ψ2 + ψ)v
and we still assume that (3.2) holds.
3.4 Extension to multidimension, multidimensional consistency
of idolons of IIII
The system (3.2) can be extended to multidimension. We denote by si (mind superscript!)
function given on edges in i-th direction of the Zn lattice, by subscript we denote forward shift
in indicated direction. The extension reads
(IIII) : sij = sj
pisi − pjsj
pjsi − pisj
, i, j = 1, . . . , n, i 6= j, (3.10)
where pi is given function and can depend only on i-th independent variable.
The crucial fact is the system is compatible
sijk = sikj . (3.11)
Moreover, we have
(Tj + 1)
[
a ln si + pi
(
bsi + c
1
si
)]
= (Ti + 1)
[
a ln sj + pi
(
bsj + c
1
si
)]
. (3.12)
It means that there exists scalar function ψ such that
a ln si + pi
(
bsi + c
1
si
)
= ψi + ψ, i = 1, . . . , n. (3.13)
From (3.10) and (3.13) we infer that
ψij = ψ + a ln
pisi − pjsj
pjsi − pisj
+ [(pi)2 − (pj)2]
[
b
sisj
pjsi − pisj
− c 1
pisi − pjsj
]
, (3.14)
i, j = 1, . . . , n, i 6= j,
where si and sj are given implicitly by means of (3.13). Due to (3.11) the system (3.14) is
multidimensionaly consistent (compatible) and we clarify what we mean by that in the following
theorem (by i-th initial line we understand in what follows the set li = {(m1, . . . ,mn) ∈ Zn | ∀ k 6=
i : mk = 0} and by set of initial lines we mean l = l1 ∪ · · · ∪ ln)
Theorem 1. For arbitrary initial condition on initial lines ψ(l) there exists solution (we do not
exclude singularities) ψ of the multidimensional system (3.13), (3.14) that obeys (3.10).
Families of Integrable Equations 9
Proof. Indeed, take arbitrary initial condition on initial lines ψ(l). Then choose a solution si(li)
(in general the value of si is given on the edge between vertices that ψ and ψi are given on) of
the equation
a ln si(li) + pi
(
bsi(li) + c
1
si(li)
)
= ψi(l
i) + ψ(li), i = 1, . . . , n (3.15)
(this is a place when non-uniqueness may enter). We treat si(l) as initial conditions for the
system (3.10). Due to (3.11) the solution si of (3.10) with initial conditions si(l) exists (we
admit singularities that come from zeroes of pjsi − pisj). Now, due to identity (3.12) there
exists function ψ such that (3.13) holds and the value of ψ at the intersection of initial lines is
equal to initial condition at the intersection of initial lines. Since si obeys (3.10) ψ satisfies (3.14)
as well. Finally ψ satisfies the assumed arbitrary initial condition since formulas (3.13) at initial
lines coincides with (3.15). �
We refer to the system (3.10) as to n-dimensional Idea system III and that is why we have
denoted it by IIII.
4 Maps
As we have already mentioned our inspiration was a survey on Yang–Baxter maps. Our goal
now is to relate our findings to some results of the papers [12, 13] and justify why it makes sense
to talk about the Idea systems
sij = sj
pisi − pjsj
pjsi − pisj
, i = 1, . . . , n (4.1)
associated with maps of type III rather than single Idea system. The Idea systems are related
by point transformation.
Indeed, first we perform a cosmetic point transformation si = pivi, pi
2 → pi and we get
vij =
vj
pi
pivi − pjvj
vi − vj
,
which in two-dimensional case after identification analogous to the one showed on the Fig. 1
yields FIII map of paper [12]
(FIII) : U =
v
p
pu− qv
u− v
, V =
u
q
pu− qv
u− v
. (4.2)
In fact by FIII we understand equivalence class of Yang–Baxter maps (cf. [13]) the equations (4.1)
and (4.2) belongs to.
Now after the point transformation vi = ui(−1)m1+···+mn we get
uij = −u
j
pi
piui − pjuj
ui − uj
associated 2D map of which is
(cHA
III) : U = −v
p
pu− qv
u− v
, V = −u
q
pu− qv
u− v
. (4.3)
After another point transformation ui = wi
(−1)m1+···+mn
pi
1
2
[(−1)m1+···+mn−1]
we obtain
wij = − 1
wj
wi − wj
piwi − pjwj
10 P. Kassotakis and M. Nieszporski
and its associated map
(cHB
III) : U = −1
v
u− v
pu− qv
, V = −1
u
u− v
pu− qv
. (4.4)
Maps (4.3) and (4.4) are not Yang–Baxter maps but they are companions (if f : (u, v) 7→ (U, V ) is
involutive map then the map (u, V ) 7→ (U, v) we refer to as companion of map f , cf. [12]) of Yang–
Baxter maps HA
III, H
B
III of paper [17]. The maps HA
III, H
B
III can be obtained in two-dimensional
case by the point transformation u1 = x, u2 = −y and w1 = x, w2 = − 1
qy respectively
x2 =
y
p
px+ qy
x+ y
, y1 =
x
q
px+ qy
x+ y
and x2 = y
qxy + 1
pxy + 1
, y1 = x
pxy + 1
qxy + 1
and then by mentioned identification (see Fig. 1)
(HA
III) : U =
v
p
pu− qv
u− v
, V =
u
q
pu− qv
u− v
,
(HB
III) : U = v
quv + 1
puv + 1
, V = u
quv + 1
puv + 1
.
Idea systems (HA
III) and (HB
III) cannot be extended to multidimension (in the sense of [25]).
Finally, we list in the Table 1 basic identities of the maps that leads to existence of potentials
of the Idea systems to illustrate how the basis changes when one changes a map.
Table 1. Basic identities of the maps that leads to existence of potentials of the Idea system.
Type of the map Example of the map Identities
U =
v
p
pu− qv
u− v
U
V
=
qv
pu
FIII pU − qV = −(pu− qv)
V =
u
q
pu− qv
u− v
1
U
− 1
V
= −
(
1
u
− 1
v
)
U = −v
p
pu− qv
u− v
U
V
=
qv
pu
cHA
III pU − qV = pu− qv
V = −u
q
pu− qv
u− v
1
U
− 1
V
=
1
u
− 1
v
U =
1
v
u− v
qv − pu
U
V
=
u
v
cHB
III pU +
1
U
− qV − 1
V
= pu+
1
u
− qv − 1
v
V =
1
u
u− v
qv − pu
pU − 1
U
− qV +
1
V
= −
(
pu− 1
u
− qv +
1
v
)
U =
v(pu+ qv)
p(u+ v)
U
V
=
qv
pu
HA
III pU + qV = pu+ qv
V =
u(pu+ qv)
q(u+ v)
1
U
+
1
V
=
1
u
+
1
v
U = v
quv + 1
puv + 1
UV = uv
HB
III pU + qV +
1
U
+
1
V
= pu+ qv +
1
u
+
1
v
V = u
puv + 1
quv + 1
pU − qV − 1
U
+
1
V
= −
(
pu− qv − 1
u
+
1
v
)
Families of Integrable Equations 11
5 Hirota’s KdV lattice equation
As the second example we consider Hirota’s KdV lattice equation [5]
x12 − x = κ
(
1
x2
− 1
x1
)
.
By the substitution u = x1x, v = x2x, we get
u2 = v + κ
(
1− v
u
)
, v1 = u+ κ
(
−1 +
u
v
)
. (5.1)
On applying identification (3.3)
u = u(m,n), v = v(m,n), U = u(m,n+ 1), V = v(m+ 1, n) (5.2)
we obtain an involutive mapping associated to system (5.1)
U = v + κ
(
1− v
u
)
, V = u+ κ
(
−1 +
u
v
)
. (5.3)
Mapping (5.3) satisfies (this is the outcome of searching for such functions F and G that F (U)+
G(V ) = f(u) + g(v) as described in the previous section):
U
V
=
v
u
, (U − κ)(V + κ) = (u− κ)(v + κ),
V (U − κ)
U(V + κ)
=
v(u− κ)
u(v + κ)
,
hence (coming back to lattice variables (5.2)) we can introduce the potentials x, y and z
u = x1x, v = x2x,
u− κ = y1/y, v + κ = y/y2,
u− κ
u
= z1/z,
v + κ
v
= z2/z. (5.4)
Eliminating u and v from (5.1) we arrive at the following lattice equations
x12 − x = κ(1/x2 − 1/x1), y1y − y12y1 = κ(y12y + y1y2),
z12z + z1z2 = z12z2 + z12z1. (5.5)
One can treat the equations as representatives of a three-parameter family of equations on φ
φ12φ
φ1φ2
=
[
(u− κ)(v + κ) + κ2
]a(−1)m+n+1−b
ub−cvb+c,
φ1
φ
= ua(−1)
m+n−b(u− κ)b+c,
φ2
φ
= va(−1)
m+n−b(v + κ)b−c, (5.6)
corresponding to the choice of parameters b = 0 = c, a = 0 = b and a = 0 = c respectively.
What more important is that from (5.4) we infer
z1
z
=
y1
x1xy
,
z2
z
=
y
y2x2x
.
Compatibility condition that guarantees existence of function z reads(
x2
x1
)2
=
(
y12y
y1y2
)2
,
from where we get
x =
τ12τ
τ1τ2
, y =
τ2
τ1
, z =
τ
τ12
.
Eliminating x, y and z from (5.4) we arrive at a compatible pair of bilinear forms of Hirota’s
KdV (cf. [19])
τ112τ − κτ11τ2 = τ12τ1, τ122τ + κτ22τ1 = τ12τ2.
12 P. Kassotakis and M. Nieszporski
6 Bäcklund transformations between idolons
In both presented examples one can find Bäcklund transformation between idolons. For instance
eliminating u and v from first two lines of (5.4) one gets Bäcklund transformation between first
two equations of (5.5)
y1
y
= x1x− k,
y
y2
= x2x+ k.
Similarly in the case of IIII one can obtain Bäcklund transformation (2.3).
Finally, we present the Bäcklund transformation between A10 (2.2) and the idolon (3.8), (3.9).
Namely, if y satisfies A10 then
• function ψ given by
a ln
p
y1 + y
+
bp2
y1 + y
+ c(y1 + y) = ψ1 + ψ,
a ln
q
y2 + y
+
bq2
y2 + y
+ c(y2 + y) = ψ2 + ψ
exists (compatibility conditions are satisfied due to the fact that y satisfies A10);
• functions u and v given by u = p
y1+y
, v = q
y2+y
obey Idea system (3.2);
• function ψ obeys (3.8), (3.9).
7 Conclusions
In this paper we focused on two 3-parameter families of lattice equations. The first one (1.1), (1.2)
and (1.3) is related to mappings of type III which were introduced in [12, 13]. Two members
(idolons) of the later are, the Hirota’s sine-Gordon equation and the lattice Schwarzian KdV [28]
in a disguise. Generally, all idolons are connected through Bäcklund transformations and they
are 3D-consistent in the sense described in the paper. In the not-too-distant future we are
going to investigate families of equations related to given integrable systems not only by discrete
quadratures but also by Bäcklund transformation from the Definition 5.
The second family described by (5.6) and (5.1) is not 3D-consistent. Nevertheless, all of its
idolons are connected through Bäcklund transformations, and since an idolon of this family is the
Hirota’s KdV equation, the whole family inherits some integrability properties e.g. τ -function
formulation.
We would like to emphasize once more that the main object under consideration are Idea
systems (3.2) (or its n-dimensional version (3.10)) and (5.1). The main observation is that the
Idea systems admit three-dimensional vector space of scalar potentials (formulas (3.8) in case of
two-dimensional Idea IIII and (3.13) in the n-dimensional case, see also second and third formulas
of (5.6)). In a forthcoming paper we will discuss all Idea systems that arise from equations of
Adler–Bobenko–Suris list. In other words, we plan to investigate all mappings in [12, 13],
determine their associate Idea systems and put more light into integrability properties of the
associated family of lattice equations. Also, it will be interesting to investigate the mappings
that arise when one imposes periodic staircase initial data on these families of lattice equations.
Another objective is to derive the discrete Painlevé equations associated with these families.
Finally, we will discuss the case of real-valued functions, which can lead to standard 3D-
consistent lattice equations. For instance for the idolon we proposed in [18]
f12 = f + (p− q)
[
v − u+
f1 − f2
(u− v)2
+
(p− q)2
(u− v)3
]
, (7.1)
Families of Integrable Equations 13
u3 + au = f1 − f, v3 + bv = f2 − f, a− b = 3(q − p).
assuming f : Z2 → R and a, b > 0 the only real solutions of the cubic equations are
u =
3
√
f1 − f
2
+
√
(f1 − f)2
4
+
a3
27
+
3
√
f1 − f
2
−
√
(f1 − f)2
4
+
a3
27
,
v =
3
√
f2 − f
2
+
√
(f2 − f)2
4
+
b3
27
+
3
√
f2 − f
2
−
√
(f2 − f)2
4
+
b3
27
. (7.2)
Then the real lattice equation (7.1), with u and v given by (7.2), is 3D-consistent.
From another perspective, instead of dealing with the family of lattice equations, it seems
more fundamental to define a model that consists of the Idea system and the associate potential
equation (e.g. equations (3.2), (3.7) or (3.10), (3.13) for the multidimensional extension). Then
the family of 3D-consistent (see Theorem 1) lattice equations follows naturally. But what more
important, this is a new lattice model, defined in both vertices and edges of a 2D square lattice
(nD lattice in the multidimensional extension). Such models have also been introduced in the
recent work of Hietarinta and Viallet [33].
Acknowledgements
We would like to thank organizers of SIDE-9 conference in Varna for their hospitality and
financial support. Special thanks to Georgi Grahovski for showing us the other side of Varna.
M.N. thanks Frank Nijhoff for pointing papers [20, 21].
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1 Introduction
2 Point transformations, difference substitutions, Bäcklund transformations and equivalence of lattice equations
3 Outline of the method
3.1 From equations to involutive maps. Idea system
3.2 Finding functions such that F(U)+G(V)=f(u)+g(v)
3.3 Potentials of the Idea systems. Idolons
3.4 Extension to multidimension, multidimensional consistency of idolons of IIII
4 Maps
5 Hirota's KdV lattice equation
6 Bäcklund transformations between idolons
7 Conclusions
References
|
| id | nasplib_isofts_kiev_ua-123456789-147991 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T08:13:06Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kassotakis, P. Nieszporski, M. 2019-02-16T12:46:22Z 2019-02-16T12:46:22Z 2011 Families of Integrable Equations / P. Kassotakis, M. Nieszporski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B20; 37K35; 39A05 http://dx.doi.org/10.3842/SIGMA.2011.100 https://nasplib.isofts.kiev.ua/handle/123456789/147991 We present a method to obtain families of lattice equations. Specifically we focus on two of such families, which include 3-parameters and their members are connected through Bäcklund transformations. At least one of the members of each family is integrable, hence the whole family inherits some integrability properties. This paper is a contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)” (June 14–18, 2010, Varna, Bulgaria). The full collection is available at http://www.emis.de/journals/SIGMA/SIDE-9.html. We would like to thank organizers of SIDE-9 conference in Varna for their hospitality and financial support. Special thanks to Georgi Grahovski for showing us the other side of Varna. M.N. thanks Frank Nijhof f for pointing papers [20, 21]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Families of Integrable Equations Article published earlier |
| spellingShingle | Families of Integrable Equations Kassotakis, P. Nieszporski, M. |
| title | Families of Integrable Equations |
| title_full | Families of Integrable Equations |
| title_fullStr | Families of Integrable Equations |
| title_full_unstemmed | Families of Integrable Equations |
| title_short | Families of Integrable Equations |
| title_sort | families of integrable equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147991 |
| work_keys_str_mv | AT kassotakisp familiesofintegrableequations AT nieszporskim familiesofintegrableequations |