Non-Point Invertible Transformations and Integrability of Partial Difference Equations
This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1)), where the map (w,z)→(φ(w,z),ψ(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1)) relates any of such equations to a quad-graph equation....
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| Цитувати: | Non-Point Invertible Transformations and Integrability of Partial Difference Equations / S.Y. Startsev // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. |
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Startsev, S.Y. 2019-02-10T10:33:27Z 2019-02-10T10:33:27Z 2014 Non-Point Invertible Transformations and Integrability of Partial Difference Equations / S.Y. Startsev // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A14; 37K05; 37K10; 37K35 DOI:10.3842/SIGMA.2014.066 https://nasplib.isofts.kiev.ua/handle/123456789/146625 This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1)), where the map (w,z)→(φ(w,z),ψ(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1)) relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the j-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the j-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract. The author thanks the referees for useful suggestions. This work is partially supported by the Russian Foundation for Basic Research (grant number 13-01-00070-a). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Point Invertible Transformations and Integrability of Partial Difference Equations Article published earlier |
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Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
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Non-Point Invertible Transformations and Integrability of Partial Difference Equations Startsev, S.Y. |
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Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_full |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
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Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
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Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
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non-point invertible transformations and integrability of partial difference equations |
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Startsev, S.Y. |
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Startsev, S.Y. |
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2014 |
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English |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
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This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1)), where the map (w,z)→(φ(w,z),ψ(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1)) relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the j-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the j-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract.
|
| issn |
1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/146625 |
| citation_txt |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations / S.Y. Startsev // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. |
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2025-11-24T06:30:35Z |
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2025-11-24T06:30:35Z |
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1850843063601070080 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 066, 13 pages
Non-Point Invertible Transformations
and Integrability of Partial Difference Equations
Sergey Ya. STARTSEV
Ufa Institute of Mathematics, Russian Academy of Sciences,
112 Chernyshevsky Str., Ufa, 450077, Russia
E-mail: startsev@anrb.ru
Received November 10, 2013, in final form June 11, 2014; Published online June 17, 2014
http://dx.doi.org/10.3842/SIGMA.2014.066
Abstract. This article is devoted to the partial difference quad-graph equations that can
be represented in the form ϕ(u(i+1, j), u(i+1, j+1)) = ψ(u(i, j), u(i, j+1)), where the map
(w, z) → (ϕ(w, z), ψ(w, z)) is injective. The transformation v(i, j) = ϕ(u(i, j), u(i, j + 1))
relates any of such equations to a quad-graph equation. It is proved that this transformation
maps Darboux integrable equations of the above form into Darboux integrable equations
again and decreases the orders of the transformed integrals by one in the j-direction. As an
application of this fact, the Darboux integrable equations possessing integrals of the second
order in the j-direction are described under an additional assumption. The transformation
also maps symmetries of the original equations into symmetries of the transformed equa-
tions (i.e. preserves the integrability in the sense of the symmetry approach) and acts as
a difference substitution for symmetries of a special form. The latter fact allows us to derive
necessary conditions of Darboux integrability for the equations defined in the first sentence
of the abstract.
Key words: quad-graph equation; non-point transformation; Darboux integrability; higher
symmetry; difference substitution; discrete Liouville equation
2010 Mathematics Subject Classification: 39A14; 37K05; 37K10; 37K35
1 Introduction and basic definitions
Let u be a function of integers i and j. By up,q denote the shifted value u(i + p, j + q) of this
function (in particular, u := u0,0 = u(i, j)). Let F (u, u1,0, u0,1) be a single-valued function.
Integrable (in various senses) equations of the form
u1,1 = F (u, u1,0, u0,1) (1.1)
are actively studied in recent years (see, e.g., [1, 9, 10, 11, 15] and references within). In the
present article we mostly consider so-called Darboux integrable equations, i.e. equations (1.1) for
which there exist functions I and J such that the relations
I(u0,1, u1,1, . . . , um,1) = I(u, u1,0, . . . , um,0), (1.2)
J(u1,0, u1,1, . . . , u1,n) = J(u, u0,1, . . . , u0,n) (1.3)
hold true for any solution u of the equation. (In other words, relationships (1.2), (1.3) mean
that I and J remain unchanged after the shifts in j and i, respectively.) The functions I
and J are respectively called an i-integral of order m and a j-integral of order n. The discrete
wave equation u1,1 = u1,0 + u0,1 − u gives us the simplest example of a Darboux integrable
equation. Here I = u1,0− u and J = u0,1− u. It should be noted that this work (in contrast to,
for instance, [5]) deals only with the autonomous integrals, i.e. with the integrals that do not
depend explicitly on the discrete variables i and j.
mailto:startsev@anrb.ru
http://dx.doi.org/10.3842/SIGMA.2014.066
2 S.Ya. Startsev
The equations (1.1) can be considered as difference analogues of the partial differential equa-
tions
uxy = F (u, ux, uy). (1.4)
The concept of the Darboux integrability was initially introduced for partial differential equa-
tions back in the 19th century. (The more recent term C-integrability, which was offered in [4],
in some sense generalizes this concept.) Searching for Darboux integrable equations of the
form (1.4) was started in classical works such as [6], and the most recent and complete clas-
sification result was obtained in [18] more than a century later. At present, similar classification
results are absent for Darboux integrable equations (1.1), only separate examples (see, for in-
stance, [5, 7, 12]) and the description [14] of a special case with n = 1 in (1.3) are known.
Therefore, it seems reasonable to consider a subclass of Darboux integrable equations (1.1) as
an intermediate goal.
To define this subclass, in the present paper we assume that the right-hand side of (1.1)
satisfies the inequalities
∂F
∂u
6= 0,
∂F
∂u1,0
6= 0,
∂F
∂u0,1
6= 0 (1.5)
and a relationship of the form
ϕ (u1,0, F (u, u1,0, u0,1)) = ψ(u, u0,1), (1.6)
where ϕ(w, z) and ψ(w, z) are functionally independent. It is easy to see that the left-hand side
of (1.6) is the shift of ϕ(u, u0,1) in i by virtue of (1.1). If we denote the operator of this shift
by Ti, then (1.6) reads as
Ti(ϕ(u, u0,1)) = ψ(u, u0,1),
∂ϕ
∂u
∂ψ
∂u0,1
− ∂ϕ
∂u0,1
∂ψ
∂u
6= 0. (1.7)
Thus, we consider formal difference analogues of the quasilinear partial differential equations
uxy = a(u, uy)ux + b(u, uy) (because any of these differential equations can be represented as
Dx(ϕ(u, uy)) = ψ(u, uy) in terms of the total derivative Dx). For shorter formulations, below
we refer to equation (1.7) for designating an equation of the form (1.1) that satisfies (1.6).
In a part of the article we use the stronger assumptions that the map (w, z)→(ϕ(w, z), ψ(w, z))
is injective, the integral J has second order and ϕ(u, u0,1) is uniquely expressed in terms of J
and ϕ(u0,1, u0,2). (Since we prove below that j-integrals are expressed in terms of ϕ(u, u0,1) and
its shifts in j, the last assumption is not so restrictive as it seems; moreover, it can be omitted
if a certain general statement will be proved – see the last paragraph of this section for more
details.) It should be noted that one of the most known integrable equations on the quad-graph,
the discrete Liouville equation
u1,1 =
(u1,0 − 1)(u0,1 − 1)
u
(1.8)
from [8], satisfies all of the above assumptions. Indeed, the work [2] demonstrates that
I =
(
u2,0
u1,0 − 1
+ 1
)(
u− 1
u1,0
+ 1
)
, J =
(
u0,2
u0,1 − 1
+ 1
)(
u− 1
u0,1
+ 1
)
for (1.8), the corresponding functions ϕ(w, z) = z/(w − 1), ψ(w, z) = (z − 1)/w define the
injective map and ϕ(u, u0,1) = (ϕ(u0,1, u0,2)+1)/(J−ϕ(u0,1, u0,2)−1). Thus, the present article
is devoted to ‘nearest relatives’ of the discrete Liouville equation (1.8). More precisely, we focus
Non-Point Invertible Transformations and Integrability of Partial Difference Equations 3
on interactions of a non-point invertible transformation [12] with Darboux integrability and, to
a lesser extent, higher symmetries of the quad-graph equations.
This transformation is defined in the following way. We can rewrite (1.7) in the form of the
system
v = ϕ(u, u0,1), v1,0 := Ti(v) = ψ(u, u0,1), (1.9)
express u and u0,1 in terms of v, v1,0 from (1.9) and obtain
u = Ω(v, v1,0), u0,1 = Υ(v, v1,0), (1.10)
where Ω and Υ are functionally independent. According to (1.5), the functions ϕ and ψ are
assumed here to be essentially depending on both their arguments, and (1.10) therefore implies
that Ω and Υ essentially depend on both their arguments too. The system (1.10) is equivalent
to the equation
Ω(v0,1, v1,1) = Υ(v, v1,0). (1.11)
Generally speaking, the above procedure is well-defined only locally and the right-hand sides
of (1.10) may be different for different pairs (u, u0,1). We avoid this if no more than one
pair (u, u0,1) satisfies the system (1.9) for any given v, v1,0. Under this assumption, the equa-
tion (1.11) is well-defined and the transformation v = ϕ(u, u0,1) maps all solutions of (1.7)
into solutions of (1.11). Because the set of solutions to (1.11) may be wider than the image of
solutions to (1.7) under the transformation v = ϕ(u, u0,1), we can not guarantee that the inverse
transformation u = Ω(v, v1,0) maps any solution of (1.11) into a solution of (1.7) (some pairs
(v, v1,0) may generate another equations of the form (1.7) when we perform the above procedure
in the inverse order). But we can restore all equations related to (1.11) by using the inverse
procedure. The described transformation is the direct analogue of that was offered in [16] for
differential-difference equations. Some applications of the transformation (1.9)–(1.11) can also
be found in [12, 17].
It is almost obvious that this transformation preserves Darboux integrability. But a formal
proof of this fact is still needed to demonstrate, for example, that no j-integral of (1.7) becomes
constant after substituting u = Ω(v, v1,0) into it. Such a proof is given in Section 2. More
precisely, we prove that equation (1.11) has a j-integral of order n− 1 and an i-integral of order
m+ 1 if equation (1.7) possesses j- and i-integrals of orders n and m, respectively. This reduces
the classification problem for Darboux integrable equations (1.7) admitting n-th order j-integrals
to the classification of equations (1.11) possessing j-integrals of order n− 1. As an example of
such kind, in Section 2 we completely describe the Darboux integrable equations (1.7) admitting
a second-order j-integral such that ϕ(u, u0,1) is uniquely expressed in terms of this integral and
ϕ(u0,1, u0,2).
In Section 3 we study the interaction between the transformation (1.9)–(1.11) and symme-
tries of the quad-graph equations. Analogically to the case of semi-discrete equations [16], the
transformation v = ϕ(u, u0,1) defines a difference substitution for special symmetries of (1.7)
and, under additional assumptions, maps any higher symmetry of (1.7) into a higher symmetry
of (1.11) (i.e. the transformation preserves integrability in the sense of the symmetry test1). The
former fact allows us to obtain necessary conditions of Darboux integrability for the equations
of the form (1.7) if u is uniquely expressed in terms of ϕ(u, u0,1) and u0,1.
Now, let us introduce notation and more formal definitions.
Due to the conditions (1.5), we can express any argument of the right-hand side F of (1.1)
in terms of the others and rewrite this equation, after appropriate shifts in i and j, in any of
1This test is described, for example, in [5, 10].
4 S.Ya. Startsev
the following forms
u−1,−1 = F (u, u−1,0, u0,−1), (1.12)
u1,−1 = F̂ (u, u1,0, u0,−1), (1.13)
u−1,1 = F̃ (u, u−1,0, u0,1). (1.14)
These formulas (and their consequences derived by shifts in i and j) allow us to express any
‘mixed shift’ up,q, pq 6= 0, in terms of uk,0, u0,l at least locally (i.e. in an enough small neigh-
borhood of any arbitrary selected solution of (1.1), which is considered in a finite number of
the points (i, j)). Therefore, we can formulate our reasonings only in terms of an arbitrary
solution u to (1.1) and its ‘canonical shifts’ uk,0, u0,l, k, l ∈ Z. These ‘canonical shifts’ are
called dynamical variables and can be considered as functionally independent. (The mixed shift
elimination procedure and the dynamical variables are described in more details, for example,
in [11].) We use the notation g[u] to designate that the function g depends on a finite number
of the dynamical variables. All functions are assumed to be analytical in this paper, and our
considerations are local.
In general, the right-hand sides of (1.12)–(1.14) are not uniquely defined (may vary with i, j
and with a solution in a neighborhood of which we consider these functions). This does not
matter if we use (1.12)–(1.14) to only estimate what dynamical variables do local expressions
for up,q depend on. But this is important in certain cases, and our statements therefore contain
the single-valuedness assumptions for F̂ when it is needed (to avoid difficulties discussed, for
example, in [3]).
Let Ti and Tj denote the operators of the forward shifts in i and j by virtue of the equa-
tion (1.1). For any function f , they satisfy the rules
Ti(f(a, b, c, . . . )) = f(Ti(a), Ti(b), Ti(c), . . . ),
Tj(f(a, b, c, . . . )) = f(Tj(a), Tj(b), Tj(c), . . . ).
In addition, Ti and Tj map up,q into up+1,q and up,q+1, respectively, and then replace any mixed
shift of u with its expression in terms of the dynamical variables. For example,
Ti(u0,n) = Tn−1j (F ), Ti(u0,−n) = T 1−n
j (F̂ ), n ∈ N. (1.15)
Here a shift operator with a superscript k designates the k-fold application of this operator (e.g.
T 3
j := Tj ◦ Tj ◦ Tj, T−2i := T−1i ◦ T−1i and any operator with the zero superscript is the identity
mapping). The inverse (backward) shift operators T−1i and T−1j are defined in a similar way.
Definition 1. Let functions I[u] and J [u] satisfy the relations Tj(I) = I and Ti(J) = J for an
equation of the form (1.1), and let each of the functions essentially depend on at least one of the
dynamical variables. Then I[u] and J [u] are respectively called an i-integral and a j-integral of
the equation (1.1), and this equation is called Darboux integrable.
It is easy to prove that the i- and j-integrals have the form I(uk,0, uk+1,0, . . . , uk+m,0) and
J(u0,l, u0,l+1, . . . , u0,l+n), respectively (see, for example, [14]). The numbers m and n are called
order of the corresponding integral if Iuk,0Iuk+m,0 6= 0 and Ju0,lJu0,l+n 6= 0. We can set k =
l = 0 without loss of generality because T−ki and T−lj respectively map any i- and j-integrals
into i- and j-integrals again. Thus, equations (1.12)–(1.14) are in fact not needed for the above
definition.
Definition 2. An equation ut = f [u] is called a symmetry of equation (1.1) if the relation
L(f) = 0 holds true, where
L = TiTj −
∂F
∂u1,0
Ti −
∂F
∂u0,1
Tj −
∂F
∂u
.
Non-Point Invertible Transformations and Integrability of Partial Difference Equations 5
According to [2, 14], if equation (1.1) is Darboux integrable and uniquely solvable for u1,0
(i.e. the right-hand side F̂ of (1.13) is uniquely defined), and J [u] denotes its j-integral, then
there exists an operator
R =
r∑
q=0
λq(u0,%, u0,%+1, . . . , u0,s)T
q
j , λr 6= 0, (1.16)
such that
ut = R
(
η(T pj (J), T p+1
j (J), T p+2
j (J), . . . )
)
(1.17)
is a symmetry of this equation for any integer p and any function η depending on a finite number
of the arguments. Both the present paper and the article [14] (results of which we use below) in
fact describe equations that admit j-integrals and symmetries of the form (1.17) (the existence
of i-integrals is not really used in the most part of the reasonings). We therefore make additional
assumptions on a j-integral in Proposition 1 and F̂ in Corollaries 1, 2 to guarantee that the
transformed and the original equations are uniquely solvable for v1,0 and u1,0, respectively.
Proposition 1 and Corollaries 1, 2 will remain valid without these assumptions if the existence
of symmetries (1.17) for the Darboux integrable equations is proved without employing the
single-valuedness of F̂ .
2 Transformation of integrals
It is convenient for further reasoning to prove the following proposition first.
Lemma 1. All functions in the set {T pi (ϕ), T qj (ϕ) | p, q ∈ Z, q 6= 0} are functionally independent
for any equation of the form (1.7).
Proof. The function ψ can be rewritten as η1(ϕ, u), where η1 must depend on its second
argument due to the functional independence of ϕ and ψ. Using the formula Ti(ϕ) = ψ = η1(ϕ, u)
and induction on p, we obtain that T pi (ϕ) = ηp(ϕ, u, u1,0, . . . , up−1,0) depends on up−1,0 for any
p > 1. Also, ϕ can be represented as a function η−1(ψ, u) that depends on its second argument.
Therefore, T−1i (ϕ) = η−1(ϕ, u−1,0) and T pi (ϕ) = ηp(ϕ, u−1,0, . . . , up,0) depends on up,0 for p < 0.
Thus, the functions T pi (ϕ) are functionally independent because ϕ and Ti(ϕ) = ψ are functionally
independent and any other T pi (ϕ) depends on the variable that is absent in either all previous
(if p > 1) or all next (if p < 0) members of the sequence T si (ϕ).
If q > 0, then the function T qj (ϕ) = ϕ(u0,q, u0,q+1) can not be expressed in terms of T pi (ϕ),
p ∈ Z, and T sj (ϕ), s < q, because they do not depend on u0,q+1. If q < 0, then the function
T qj (ϕ) = ϕ(u0,q, u0,q+1) can not be expressed in terms of T pi (ϕ), p ∈ Z, and T sj (ϕ), s > q,
because they do not depend on u0,q. Thus, {T pi (ϕ), T qj (ϕ) | p, q ∈ Z, q 6= 0} is a set of functionally
independent functions. �
Lemma 2. Up to shifts in j, any n-th order j-integral of equation (1.7) can be represented in
the form Φ(ϕ(u, u0,1), ϕ(u0,1, u0,2), . . . , ϕ(u0,n−1, u0,n)). This representation is well-defined on
solutions of (1.1), and the function Φ essentially depends on its first and last arguments.
Proof. As it is demonstrated in the comments to Definition 1, we can assume without loss of
generality that the j-integral has the form J(u, u0,1, . . . , u0,n). Equation (1.6) implies that the
right-hand side of (1.1) has the form g(ψ(u, u0,1), u1,0), where g is single-valued. The backward
shift in i gives us u0,1 = g(ϕ(u, u0,1), u). Using this expression and its consequences derived
6 S.Ya. Startsev
by shifts in j, we rewrite the j-integral as J = Φ(u, ϕ(u, u0,1), ϕ(u0,1, u0,2), . . . , ϕ(u0,n−1, u0,n)).
Since
Ti
(
Φ(u, ϕ, . . . , Tn−1j (ϕ))
)
= Φ
(
u1,0, ψ, . . . , T
n−1
j (ψ)
)
by virtue of (1.7), the relation Ti(Φ) = Φ can hold true only if Φ does not depend on its first
argument (Φu = 0). The function Φ also must depends on ϕ and Tn−1j (ϕ) because Ju = 0 and
Ju0,n = 0 otherwise. �
Theorem 1. Let the map (w, z)→ (ϕ(w, z), ψ(w, z)) be injective, and the corresponding equa-
tion (1.11) be uniquely solvable for v1,1. Then the equation (1.7) possesses an i-integral of
order m and a j-integral of order n if and only if the equation (1.11) possesses an i-integral of
order m+ 1 and a j-integral of order n− 1. If n = 2 and (1.11) is not uniquely solvable for v1,1,
then the Darboux integrability of (1.7) implies the existence of an (m + 1)th order i-integral
and a first-order j-integral for an equation v1,1 = Q(v, v1,0, v0,1) that satisfies the relationship
Ω(v0,1, Q(v, v1,0, v0,1)) = Υ(v, v1,0).
Proof. We can assume without loss of generality that the integrals of (1.7) have the form
I(u, u1,0, . . . , um,0) and J(u, u0,1, . . . , u0,n). Substituting Ω(ϕ, Ti(ϕ)) instead of u into the i-
integral, we obtain I(Ω,Ω1, . . . ,Ωm), where Ωk = Ω(T ki (ϕ), T k+1
i (ϕ)). The relationship Tj(I) = I
takes the form
I(Υ,Υ1, . . . ,Υm) = I(Ω,Ω1, . . . ,Ωm), Υk = Υ
(
T ki (ϕ), T k+1
i (ϕ)
)
, (2.1)
because v = ϕ(u, u1,0) satisfies equation (1.11) and hence Tj maps Ω(ϕ, Ti(ϕ)) into Υ(ϕ, Ti(ϕ)).
But ϕ, Ti(ϕ), . . . , Tm+1
i (ϕ) are functionally independent by Lemma 1, and (2.1) can be valid
only if the relation
I(Υ(v, v1,0),Υ(v1,0, v2,0), . . . ,Υ(vm,0, vm+1,0))
= I(Ω(v, v1,0),Ω(v1,0, v2,0), . . . ,Ω(vm,0, vm+1,0))
holds true identically for arbitrary v, v1,0, . . . , vm+1,0. And this means that the right-hand side
of the last relationship is an i-integral of equation (1.11).
According to Lemma 2, J = Φ(ϕ, . . . , Tn−1j (ϕ)). Let us rewrite (1.11) as v1,1 = Q(v, v1,0, v0,1).
Since Ti(Φ) = Ti(J) = J = Φ and equation (1.11) holds true for v = ϕ(u, u0,1), the function
Φ(v, v0,1, . . . , v0,n−1) satisfies the defining relation
Φ
(
v1,0, Q, . . . , T
n−2
j (Q)
)
= Φ(v, v0,1, . . . , v0,n−1) (2.2)
for j-integral of (1.11) when v1,0 = Ti(ϕ) and v0,k = T kj (ϕ), k = 0, n− 1 (other variables are
absent in the defining relation). But Ti(ϕ) = ψ and T kj (ϕ) are functionally independent and,
therefore, (2.2) is valid only if it holds true identically for arbitrary v1,0 and v0,k, k = 0, n− 1.
Thus, Φ(v, v0,1, . . . , v0,n−1) is a j-integral of equation (1.11).
Conversely, let I(Ω(v, v1,0),Ω(v1,0, v2,0), . . . ,Ω(vm,0, vm+1,0)) and Φ(v, v0,1, . . . , v0,n−1) be inte-
grals of (1.11). Since v = ϕ(u, u1,0) satisfies (1.11) for any solution u of the equation (1.7) and
u = Ω(ϕ, Ti(ϕ)) identically holds true, the relationships
Ti
(
Φ(ϕ, Tj(ϕ), . . . , Tn−1j (ϕ))
)
= Φ
(
ϕ, Tj(ϕ) . . . , Tn−1j (ϕ)
)
,
Tj(I(u, u1,0, . . . , um,0)) = I(u, u1,0, . . . , um,0)
follow from the defining relation for integrals of (1.11). �
Non-Point Invertible Transformations and Integrability of Partial Difference Equations 7
In particular, this theorem implies that any Darboux integrable equation (1.7) admitting
a second-order j-integral can be derived from a Darboux integrable equation possessing a first-
order j-integral. But the equations of the latter kind were described (under an additional
assumption) in the recent work [14] and we only need to select the equations of the form (1.11)
among them.
Lemma 3. Let an equation
ṽ1,1 = Q(ṽ, ṽ1,0, ṽ0,1),
∂Q
∂ṽ
∂Q
∂ṽ1,0
∂Q
∂ṽ0,1
6= 0 (2.3)
be Darboux integrable, satisfy a relationships of the form Ω̃(ṽ0,1, Q(ṽ, ṽ1,0, ṽ0,1)) = Υ̃(ṽ, ṽ1,0) and
possess a first-order j-integral Φ(ṽ, ṽ0,1) such that the equation Φ(ṽ, ṽ0,1) = w̃ is uniquely solvable
for ṽ. Then a point transformation ṽ = ζ(v) relates (2.3) to an equation of the form
δD − v1,1(Av0,1 + C − δB)
v1,1 − v0,1
=
(v +B)(δv1,0 + C)−AD
v1,0 − v
, (2.4)
where the constants A, B, C, D and δ satisfy the inequalities |δA| + |C − δB| + |D| 6= 0,
|δ|+ |A|+ |C| 6= 0.
Proof. The defining relation Φ(ṽ1,0, Q) = Φ(ṽ, ṽ0,1) for the j-integral Φ is uniquely solvable for
the first argument of the left-hand side by the theorem assumptions. Hence, the equation (2.3) is
uniquely solvable for ṽ1,0. According to [14], an equation of the form (2.3) is Darboux integrable,
uniquely solvable for ṽ1,0 and admits an autonomous first-order j-integral only if a point change
of variables ṽ = ζ(v) relates this equation to an equation of the form
v1,1 = α(φ) +
β(φ)
γ(φ)− v1,0
, β 6= 0, (2.5)
where φ is the function of v and v0,1 that satisfies the relationship
v0,1 = α(φ) +
β(φ)
γ(φ)− v
. (2.6)
It is obvious that the last relationship can hold true only if |α′|+ |β′|+ |γ′| 6= 0. The function φ
is a j-integral, and
I[v] =
v3,0 − v1,0
v3,0 − v2,0
· v2,0 − v
v1,0 − v
(2.7)
is an i-integral of (2.5). It should be noted that some equations of the form (2.5), (2.6) admit i-
integrals of order less than 3, but all such equations possess the integral (2.7) too (see [14] for
more details).
An equation v1,1 = F (v, v1,0, v0,1) can be written in the form (1.11) only if the condition
(Fv/Fv1,0)v0,1 = 0 holds true. Substituting the right-hand side of (2.5) into this condition, we
obtain
∂
∂v0,1
(
α′(γ − v1,0)2 + β′(γ − v1,0)− βγ′
β
∂φ
∂v
)
= 0. (2.8)
On the other hand, differentiation of (2.6) with respect to v gives rise to
β =
(
βγ′ − α′(γ − v)2 − β′(γ − v)
)∂φ
∂v
.
8 S.Ya. Startsev
The condition β 6= 0 implies that the both factors in the right-hand side do not equal zero.
Therefore, (2.8) takes the form
∂
∂v0,1
(
α′(γ − v1,0)2 + β′(γ − v1,0)− βγ′
α′(γ − v)2 + β′(γ − v)− βγ′
)
= 0.
Performing the differentiation in the left-hand side of the last relationship, we obtain[
(α′γ′′ − α′′γ′)
(
v21,0 − 2γv1,0
)
+
(
γ′β′′ − γ′′β′ + 2α′(γ′)2
)
v1,0
]
β
+
(
α′′β′ − α′β′′ − 2(α′)2γ′
)(
v21,0(γ − v) + v1,0
(
v2 − γ2
))
+ · · · = 0,
where the dots signify terms without v1,0. The left-hand side is a polynomial in v and v1,0. The
coefficients of it depend on φ only and must be equal to zero because φ(v, v0,1), v and v1,0 are
functionally independent. Thus, we have
α′γ′′ − α′′γ′ = γ′β′′ − γ′′β′ + 2α′(γ′)2 = α′β′′ − α′′β′ + 2(α′)2γ′ = 0. (2.9)
If φ satisfies (2.6), then any function of φ also satisfies (2.6) (with another α, β and γ). This is
why we can assume without loss of generality that α = δφ if α′ 6= 0. Under this assumption, (2.9)
gives rise to γ = Aφ−B and β = D−Cφ− δφ(Aφ−B). It is easy to check that an appropriate
change v → v + λ in (2.6) reduces the two other cases α′ = 0, γ = φ and α′ = γ′ = 0, β = φ to
the same formulas for α, β and γ with δ = 0, A = 1 and δ = A = D = 0, C = −1, respectively.
Substituting these formulas into (2.6) and solving it for φ, we obtain
φ =
v0,1(v +B) +D
Av0,1 + δv + C
. (2.10)
The corresponding equation (2.5) is
v1,1 =
((v +B)(δv1,0 + C)−AD)v0,1 + δD(v1,0 − v)
A(v1,0 − v)v0,1 + (v1,0 +B)(δv + C)−AD
and can be rewritten as (2.4). �
If we perform a point transformation v = η(ṽ) in (1.11), then the corresponding system
u = Ω(η(ṽ), η(ṽ1,0)), u0,1 = Υ(η(ṽ), η(ṽ1,0))
has the solution η(ṽ) = ϕ(u, u0,1), η(ṽ1,0) = ψ(u, u0,1) that generates the unchanged equa-
tion (1.7). Thus, we can restore the original equation (1.7) after a point transformation in (1.11).
But the equation Ω̃(ṽ0,1, ṽ1,1) = Υ̃(ṽ, ṽ1,0) in the formulation of Lemma 3 may also differ
from (1.11) via a transformation Ω̃ = ξ(Ω), Υ̃ = ξ(Υ), that corresponds to the point change
ũ = ξ(u) in (1.7).
Theorem 1, the above lemmas and the reasonings of the previous paragraph mean that
the transformation from (1.11) to (1.7) for equation (2.4) gives us all, up to point transfor-
mations, Darboux integrable equations (1.7) admitting second-order j-integrals of the form
Φ(ϕ(u, u0,1), ϕ(u0,1, u0,2)), where the equation Φ(v, v0,1) = w is uniquely solvable for v. Let us
perform this transformation.
The system (1.10) for (2.4) takes the form
u =
δD − v1,0(Av + C − δB)
v1,0 − v
, (2.11)
u0,1 =
(v +B)(δv1,0 + C)−AD
v1,0 − v
. (2.12)
Non-Point Invertible Transformations and Integrability of Partial Difference Equations 9
Equation (2.11) can be rewritten in the form
v1,0 =
uv + δD
u+Av + C − δB
. (2.13)
Replacing v1,0 with the right-hand side of (2.13), we transform (2.12) into the algebraic equation
P2v
2 + P1v + P0 = 0, where
P2 = δu+Au0,1 +AC, (2.14)
P1 = (C + δB)u+ (C − δB)u0,1 + (BC −AD − δD)(A− δ) + C2, (2.15)
P0 = (BC −AD)u− δDu0,1 +BC(C − δB) +D
(
δAB −AC + δ2B
)
. (2.16)
Solving this equation for v and substituting a solution θ(u, u0,1) into (2.13), we obtain the
following classification result.
Proposition 1. Let the map (w, z)→ (ϕ(w, z), ψ(w, z)) be injective, and let the equation (1.7)
possess a second-order autonomous j-integral J such that ϕ(u, u0,1) is uniquely expressed in
terms of J and ϕ(u0,1, u0,2). Then the equation (1.7) is Darboux integrable if and only if a point
change of variables ρ(u)→ u reduces this equation to the form
Ti(θ(u, u0,1)) =
uθ(u, u0,1) + δD
Aθ(u, u0,1) + u+ C − δB
, (2.17)
where θ is a solution of the equation P2θ
2 + P1θ + P0 = 0 with the coefficients defined by
formulas (2.14)–(2.16), and the constants A, B, C, D, δ satisfy the conditions |δ|+|A|+|C| 6= 0,
|δA|+ |C − δB|+ |D| 6= 0.
Theorem 1 guaranties that (2.17) is Darboux integrable, and the proof of Theorem 1 gives
us the way to construct integrals of this equation. Substituting θ instead of v into (2.10), we
obtain a j-integral
J [u] = φ(θ, Tj(θ)) =
Tj(θ)(θ +B) +D
ATj(θ) + δθ + C
.
Equation (2.13) (and its shifts in i) allows us to represent v1,0, v2,0, v3,0 as functions of v, u,
u1,0, u2,0. Replacing vk,0 with these functions in (2.7), we derive the formula
I[u] =
(u2,0 + u1,0 + C − δB)(u1,0 + u+ C − δB)
u21,0 + u1,0(C − δB)− δAD
for an i-integral of the equation (2.17).
As it is shown in Section 1, the discrete Liouville equation (1.8) satisfies all assumptions of
Proposition 1. Therefore, it lies in the equation family (2.17) as a particular case C = −1,
B 6= 0, A = δ = 0. Another special case C = D = 0, B = −1, A 6= −1, δ = 1 gives us the
equation u1,1(u+1) = u1,0(u0,1 +1), which is contained in a list of Darboux integrable equations
in [5].
3 Transformation of symmetries
Let h∗ designate the Frechét derivative (linearization operator)
h∗ =
+∞∑
q=−∞
∂h
∂uq,0
T qi +
+∞∑
q=−∞
q 6=0
∂h
∂u0,q
T qj
10 S.Ya. Startsev
of the function h[u], and let ∂f denote the differentiation with respect to t by virtue of the
equation ut = f [u]. The formula ∂f (h[u]) = h∗(f) defines ∂f on the functions of the dynamical
variables.
Let us consider a differential-difference equation of the form
ut = g(u0,n, u0,n+1, . . . , u0,k). (3.1)
Definition 3. If a function φ(u0,l, u0,l+1, . . . , u0,m), m > l, satisfies the inequality φu0,lφu0,m 6= 0
and a relationship2 of the form
∂g(φ) = ĝ
(
Tnj (φ), Tn+1
j (φ), . . . , T kj (φ)
)
, (3.2)
then we say that equation (3.1) admits the difference substitution
v = φ(u0,l, u0,l+1, . . . , u0,m) (3.3)
into the equation vt = ĝ(v0,n, v0,n+1, . . . , v0,k).
We call (3.3) a Miura-type substitution if there exist operators (1.16) and
R̂ =
r+m∑
q=l
λ̂q(v0,%̂, v0,%̂+1, . . . , v0,ŝ)T
q
j
such that the equation ut = R(ξ(T pj (φ), T p+1
j (φ), . . . )) admits the substitution (3.3) into the
equation vt = R̂(ξ(v0,p, v0,p+1, . . . )) for any integer p and any function ξ depending on a finite
number of the arguments.
Since mixed shifts uq,p, qp 6= 0, do not appear in the defining relation (3.2), the above
definition in no way uses equation (1.1). However, equations (1.1) can generate difference
substitutions, for example, in the way that was used in [16] for differential-difference analogues
of equations (1.7).
Theorem 2. If (3.1) is a symmetry of (1.7), then the equation (3.1) admits the difference
substitutions v = ϕ(u, u0,1) and v = ψ(u, u0,1) into an equation vt = ĝ(v0,n, v0,n+1, . . . , v0,k).
If the map (w, z) → (ϕ(w, z), ψ(w, z)) is injective and the corresponding equation (1.11)
is uniquely solvable for v1,0, v1,1 and v0,1, then the following proposition is also true for any
symmetry ut = f [u] of equation (1.7) such that f 6= 0. Let f̂ [v] designate the function ϕ∗(f)
after substituting u = Ω(v, v1,0) into it and excluding the variables of the form vp,q, pq 6= 0, by
virtue of (1.11). Then f̂ [v] 6= 0 and vt = f̂ [v] is a symmetry of equation (1.11). If, in addition,
fuδ,0 6= 0 for some δ > 0 (δ < 0) or fu0,σ 6= 0 for some σ > 0 (σ < 0), then f̂ [v] essentially
depends on vd,0 for some d ≥ δ (d ≤ δ) or on v0,b for some b ≥ σ (b ≤ σ), respectively.
Although the proof is almost identical to the proof of the analogous proposition for semi-
discrete equations in [16], we include it for the sake of completeness. It should also be noted
that for the rest part of the present article we need only the first sentence of the theorem.
Proof. The application of the Frechét derivative to both sides of (1.6) gives rise to
Ti
(
∂ϕ
∂u0,1
)
F∗ + Ti
(
∂ϕ
∂u
)
= ψ∗ ⇒ Ti
(
∂ϕ
∂u0,1
)
L = Ti
(
∂ϕ
∂u0,1
)
(TiTj − F∗) = Ti ◦ ϕ∗ − ψ∗,
where ◦ denotes the composition of operators. Thus, the defining relation L(f) = 0 for the
symmetry ut = f [u] takes the form Ti(ϕ∗(f)) = ψ∗(f).
2This relationship means that v = φ[u] maps solutions of (3.1) into solutions of the equation vt = ĝ.
Non-Point Invertible Transformations and Integrability of Partial Difference Equations 11
Let (3.1) be a symmetry of (1.7). The function ϕ∗(g) can be rewritten as
ϕ∗(g) = ĝ
(
u0,n, T
n
j (ϕ), Tn+1
j (ϕ), . . . , T kj (ϕ)
)
. (3.4)
The substitution of (3.4) into the defining relation ψ∗(g) = Ti(ϕ∗(g)) gives rise to
ψ∗(g) = ĝ
(
Ti(u0,n), Tnj (ψ), Tn+1
j (ψ), . . . , T kj (ψ)
)
. (3.5)
But (1.5) and (1.15) imply that Ti(u0,n) depends on u1,0 whereas ψ∗(g) and Tnj (ψ), . . . , T kj (ψ)
do not. Therefore, ĝ does not depend on its first argument and (3.4), (3.5) take the form
ϕ∗(g) = ĝ
(
Tnj (ϕ), Tn+1
j (ϕ), . . . , T kj (ϕ)
)
,
ψ∗(g) = ĝ
(
Tnj (ψ), Tn+1
j (ψ), . . . , T kj (ψ)
)
,
respectively.
For the symmetries of the general form ut = f [u], equation (1.10) and the relationship
Ti(ϕ∗(f)) = ψ∗(f) imply
∂f (Ω(ϕ,ψ)) =
∂Ω
∂v
ϕ∗(f) +
∂Ω
∂v1,0
ψ∗(f) =
∂Ω
∂v
ϕ∗(f) +
∂Ω
∂v1,0
Ti(ϕ∗(f)) = ∂f (u) = f, (3.6)
∂f (Υ(ϕ,ψ)) =
∂Υ
∂v
ϕ∗(f) +
∂Υ
∂v1,0
ψ∗(f) =
∂Υ
∂v
ϕ∗(f) +
∂Υ
∂v1,0
Ti(ϕ∗(f)) = ∂f (u0,1) = Tj(f).
The last two formulas, together with ψ = Ti(ϕ), mean that the function f̆ [u] := ϕ∗(f) satisfies
the relationship
Tj
(
∂Ω(ϕ, Ti(ϕ))
∂v
f̆ +
∂Ω(ϕ, Ti(ϕ))
∂v1,0
Ti(f̆)
)
= Tj(f)
=
∂Υ(ϕ, Ti(ϕ))
∂v
f̆ +
∂Υ(ϕ, Ti(ϕ))
∂v1,0
Ti(f̆). (3.7)
Using the identities u = Ω(ϕ, Ti(ϕ)), Ω(Tj(ϕ), TiTj(ϕ)) = Υ(ϕ, Ti(ϕ)) and their consequences
derived by shifts in i and j, we can represent f̆ [u] as a function f̂ [ϕ] of only T pi (ϕ), T qj (ϕ),
p, q ∈ Z, q 6= 0, and exclude all mixed shifts T si T
r
j (ϕ), sr 6= 0, from (3.7). As a result of this,
in (3.7) the operators Ti and Tj act on vp,0 = T pi (ϕ), v0,q = T qj (ϕ) by virtue of (1.11), i.e. (3.7)
becomes the defining relation for symmetries of (1.11). But the functions T pi (ϕ), T qj (ϕ) are
functionally independent by Lemma 1, and f̂ [v] therefore satisfies this defining relation not
only if vp,0 = T pi (ϕ), v0,q = T qj (ϕ) but for arbitrary values of vp,0, v0,q.
If ϕ∗(f) = 0, then (3.6) implies f = 0. This is why f̂ [v] 6= 0 if f 6= 0. The same logic
proves the dependence of f̂ [v] on vd,0 if we take the relationship TiT
q
j (ϕ) = ψ(u0,q, u0,q+1) and
the formulas for T pi (ϕ), T qj (ϕ) from the proof of Lemma 1 into account. The dependence of f̂ [v]
on v0,b follows directly form the relation ϕu0,1Tj(f) + ϕuf = f̂ [v]
∣∣
v=ϕ
and the fact that f can
not depend on u0,σ for positive or negative σ if f̂ [v] does not depend on v0,b for all b ≥ σ or for
all b ≤ σ, respectively. �
Corollary 1. If an equation of the form (1.7) is uniquely solvable for u1,0 and Darboux inte-
grable, then v = ϕ(u, u0,1) and w = ψ(u, u0,1) are Miura-type substitutions.
Recall that in the present paper we understand (1.7) as an equation of the form (1.1) satisfying
the relationship (1.6). According to this, in Corollaries 1, 2 we assume that the equation (1.1) is
uniquely solvable for u1,0 (i.e. the right-hand side of the corresponding equation (1.13) is well-
defined). We can replace this assumption with the more strong assumption that the formula
v = ϕ(u, u0,1) is uniquely solvable for u. The last assumption guarantees that any equation (1.1)
satisfying (1.6) is uniquely solvable for u1,0.
12 S.Ya. Startsev
Proof. It is easy to see that R̂ in Definition 3 coincides with the operator φ∗ ◦R =
r+m∑
q=l
λ̆q[u]T qj
and v = φ[u] is a Miura-type substitution if all λ̆q[u] can be expressed in terms of T bj (φ) only.
According to [2, 14], if equation (1.1) is Darboux integrable and uniquely solvable for u1,0, then
it possesses symmetries of the form (1.16), (1.17). Theorem 2 implies that the symmetries (1.17)
admit the substitutions v = ϕ(u, u0,1) and w = ψ(u, u0,1), i.e. ϕ∗(R(η)) and ψ∗(R(η)) can be
respectively expressed in terms of v0,b = T bj (ϕ) and w0,b := T bj (ψ), b ∈ Z. But in the proof of
Lemma 2 we demonstrate that any j-integral J [u] of (1.7) can be represented both as a function
of T bj (ϕ) and as a function of T bj (ψ). Hence, the same is true for any function η of the j-integrals.
Due to the arbitrariness of η in (1.17), this implies that the coefficients of the operators ϕ∗◦R and
ψ∗ ◦R can be expressed in terms of T bj (ϕ) and T bj (ψ), respectively. Thus, the equation ut = R(ξ)
admits the substitutions v = ϕ(u, u0,1) and w = ψ(u, u0,1) for any function ξ depending on
arguments of the forms T bj (ϕ) and T bj (ψ), respectively. �
It is proved in [13] that v = φ(u, u0,1) is a Miura-type substitution only if φ satisfies the
relationship
ζ(u0,1) = α(φ) +
β(φ)
γ(φ)− ζ(u)
,
for some functions α, β, γ and ζ such that βζ ′ 6= 0. This implies the following proposition.
Corollary 2. If an equation of the form (1.7) is uniquely solvable for u1,0 and Darboux inte-
grable, then there exist functions α, β, γ, ζ and α̂, β̂, γ̂, ζ̂ such that
ζ(u0,1) = α(ϕ) +
β(ϕ)
γ(ϕ)− ζ(u)
, βζ ′ 6= 0,
ζ̂(u0,1) = α̂(ψ) +
β̂(ψ)
γ̂(ψ)− ζ̂(u)
, β̂ζ̂ ′ 6= 0.
Applying Ti and T−1i to the both sides of the former and the latter relationships, respectively,
we obtain that any Darboux integrable equation (1.7) can be represented in the form
ζ(u1,1) = α(ψ) +
β(ψ)
γ(ψ)− ζ(u1,0)
as well as in the form
ζ̂(u−1,1) = α̂(ϕ) +
β̂(ϕ)
γ̂(ϕ)− ζ̂(u−1,0)
.
We can assume either ζ(u) = u or ζ̂(u) = u without loss of generality because we can perform
either the point change of variables ζ(u)→ u or the point change ζ̂(u)→ u.
Acknowledgments
The author thanks the referees for useful suggestions. This work is partially supported by the
Russian Foundation for Basic Research (grant number 13-01-00070-a).
Non-Point Invertible Transformations and Integrability of Partial Difference Equations 13
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http://dx.doi.org/10.1007/s10688-009-0002-5
http://arxiv.org/abs/0705.1663
http://dx.doi.org/10.1007/BF02557219
http://arxiv.org/abs/solv-int/9902016
http://dx.doi.org/10.1093/imrn/rnt066
http://arxiv.org/abs/1204.0638
http://dx.doi.org/10.1088/1751-8113/45/34/345205
http://arxiv.org/abs/1203.4369
http://dx.doi.org/10.5802/afst.153
http://dx.doi.org/10.5802/afst.153
http://dx.doi.org/10.1063/1.3628587
http://arxiv.org/abs/1102.1236
http://dx.doi.org/10.1143/JPSJ.56.4285
http://dx.doi.org/10.3842/SIGMA.2011.079
http://arxiv.org/abs/1108.3648
http://dx.doi.org/10.1088/1751-8113/42/45/454012
http://arxiv.org/abs/0902.4421
http://dx.doi.org/10.1007/s11232-011-0033-y
http://arxiv.org/abs/1004.5346
http://dx.doi.org/10.3842/SIGMA.2010.092
http://arxiv.org/abs/1010.0361
http://matem.anrb.ru/en/article?art_id=182
http://dx.doi.org/10.1088/1751-8113/47/10/105204
http://arxiv.org/abs/1310.2282
http://dx.doi.org/10.1016/S0375-9601(00)00806-9
http://dx.doi.org/10.1016/S0375-9601(00)00806-9
http://arxiv.org/abs/0709.3095
http://dx.doi.org/10.1007/BF01018403
http://dx.doi.org/10.1007/BF01018403
http://dx.doi.org/10.1088/0305-4470/27/20/020
http://dx.doi.org/10.1070/rm2001v056n01ABEH000357
1 Introduction and basic definitions
2 Transformation of integrals
3 Transformation of symmetries
References
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