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 $\varphi (u(i+1,j),u(i+1,j+1))=\psi (u(i,j),u(i,j+1))$, where the map $(w,z) \rightarrow (\varphi(w,z),\psi(w,z))$ is injective. The transformation $v(i,j)=\varphi (u(i,j),u(i,j+1))$ 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.

( 1.4) The concept of the Darboux integrability was initially introduced for partial differential equations 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 classification 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 instance, [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 = 0, ∂F ∂u 1,0 = 0, ∂F ∂u 0,1 = 0 (1.5) and a relationship of the form ϕ (u 1,0 , F (u, u 1,0 , u 0,1 )) = ψ(u, u 0,1 ), (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, u 0,1 ) in i by virtue of (1.1). If we denote the operator of this shift by T i , then (1.6) reads as Thus, we consider formal difference analogues of the quasilinear partial differential equations u xy = a(u, u y )u x + b(u, u y ) (because any of these differential equations can be represented as D x (ϕ(u, u y )) = ψ(u, u y ) in terms of the total derivative D x ). 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, u 0,1 ) is uniquely expressed in terms of J and ϕ(u 0,1 , u 0,2 ). (Since we prove below that j-integrals are expressed in terms of ϕ(u, u 0,1 ) 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 from [8], satisfies all of the above assumptions. Indeed, the work [2] demonstrates that 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, u 0,1 ) = (ϕ(u 0,1 , u 0,2 ) + 1)/(J − ϕ(u 0,1 , u 0,2 ) − 1). Thus, the present article is devoted to 'nearest relatives' of the discrete Liouville equation (1.8). More precisely, we focus 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, u 0,1 ), v 1,0 := T i (v) = ψ(u, u 0,1 ), (1.9) express u and u 0,1 in terms of v, v 1,0 from (1.9) and obtain 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 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, u 0,1 ). We avoid this if no more than one pair (u, u 0,1 ) satisfies the system (1.9) for any given v, v 1,0 . Under this assumption, the equation (1.11) is well-defined and the transformation v = ϕ(u, u 0,1 ) 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, u 0,1 ), we can not guarantee that the inverse transformation u = Ω(v, v 1,0 ) maps any solution of (1.11) into a solution of (1.7) (some pairs (v, v 1,0 ) 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, v 1,0 ) 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, u 0,1 ) is uniquely expressed in terms of this integral and ϕ(u 0,1 , u 0,2 ).
In Section 3 we study the interaction between the transformation (1.9)-(1.11) and symmetries of the quad-graph equations. Analogically to the case of semi-discrete equations [16], the transformation v = ϕ(u, u 0,1 ) 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 test 1 ). 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, u 0,1 ) and u 0,1 . 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 the following forms (1.14) These formulas (and their consequences derived by shifts in i and j) allow us to express any 'mixed shift' u p,q , pq = 0, in terms of u k,0 , u 0,l at least locally (i.e. in an enough small neighborhood 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' u k,0 , u 0,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 u p,q depend on. But this is important in certain cases, and our statements therefore contain the single-valuedness assumptions forF when it is needed (to avoid difficulties discussed, for example, in [3]).
Let T i and T j denote the operators of the forward shifts in i and j by virtue of the equation (1.1). For any function f , they satisfy the rules In addition, T i and T j map u p,q into u p+1,q and u p,q+1 , respectively, and then replace any mixed shift of u with its expression in terms of the dynamical variables. For example, Here a shift operator with a superscript k designates the k-fold application of this operator (e.g. It is easy to prove that the i-and j-integrals have the form I(u k,0 , u k+1,0 , . . . , u k+m,0 ) and J(u 0,l , u 0,l+1 , . . . , u 0,l+n ), respectively (see, for example, [14]). The numbers m and n are called order of the corresponding integral if I u k,0 I u k+m,0 = 0 and J u 0,l J u 0,l+n = 0. We can set k = l = 0 without loss of generality because T −k i and T −l j 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.
According to [2,14], if equation (1.1) is Darboux integrable and uniquely solvable for u 1,0 (i.e. the right-hand sideF of (1.13) is uniquely defined), and J[u] denotes its j-integral, then there exists an operator 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 andF in Corollaries 1, 2 to guarantee that the transformed and the original equations are uniquely solvable for v 1,0 and u 1,0 , 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 ofF .

Transformation of integrals
It is convenient for further reasoning to prove the following proposition first. Proof . The function ψ can be rewritten as η 1 (ϕ, u), where η 1 must depend on its second argument due to the functional independence of ϕ and ψ. Using the formula T i (ϕ) = ψ = η 1 (ϕ, u) and induction on p, we obtain that T p i (ϕ) = η p (ϕ, u, u 1,0 , . . . , u p−1,0 ) depends on u p−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 −1 i (ϕ) = η −1 (ϕ, u −1,0 ) and T p i (ϕ) = η p (ϕ, u −1,0 , . . . , u p,0 ) depends on u p,0 for p < 0. Thus, the functions T p i (ϕ) are functionally independent because ϕ and T i (ϕ) = ψ are functionally independent and any other T p i (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 s i (ϕ). If q > 0, then the function T q j (ϕ) = ϕ(u 0,q , u 0,q+1 ) can not be expressed in terms of T p i (ϕ), p ∈ Z, and T s j (ϕ), s < q, because they do not depend on u 0,q+1 . If q < 0, then the function T q j (ϕ) = ϕ(u 0,q , u 0,q+1 ) can not be expressed in terms of T p i (ϕ), p ∈ Z, and T s j (ϕ), s > q, because they do not depend on u 0,q . Thus, {T p i (ϕ), T q j (ϕ) | p, q ∈ Z, q = 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, u 0,1 ), ϕ(u 0,1 , u 0,2 ), . . . , ϕ(u 0,n−1 , u 0,n )). This representation is well-defined on solutions of (1.1), and the function Φ essentially depends on its first and last arguments.
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 firstorder 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.
Proof . The defining relation Φ(ṽ 1,0 , Q) = Φ(ṽ,ṽ 0,1 ) 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 forṽ 1,0 . According to [14], where φ is the function of v and v 0,1 that satisfies the relationship It is obvious that the last relationship can hold true only if |α | + |β | + |γ | = 0. The function φ is a j-integral, and is an i-integral of (2.5). It should be noted that some equations of the form (2.5), (2.6) admit iintegrals of order less than 3, but all such equations possess the integral (2.7) too (see [14] for more details). An equation v 1,1 = F (v, v 1,0 , v 0,1 ) can be written in the form (1.11) only if the condition (F v /F v 1,0 ) v 0,1 = 0 holds true. Substituting the right-hand side of (2.5) into this condition, we obtain On the other hand, differentiation of (2.6) with respect to v gives rise to The condition β = 0 implies that the both factors in the right-hand side do not equal zero. Therefore, (2.8) takes the form Performing the differentiation in the left-hand side of the last relationship, we obtain where the dots signify terms without v 1,0 . The left-hand side is a polynomial in v and v 1,0 . The coefficients of it depend on φ only and must be equal to zero because φ(v, v 0,1 ), v and v 1,0 are functionally independent. Thus, we have 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 = 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 The corresponding equation (2.5) is and can be rewritten as (2.4).
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 Equation (2.13) (and its shifts in i) allows us to represent v 1,0 , v 2,0 , v 3,0 as functions of v, u, u 1,0 , u 2,0 . Replacing v k,0 with these functions in (2.7), we derive the formula for an i-integral of the equation (2.17).
Although the proof is almost identical to the proof of the analogous proposition for semidiscrete 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.