On Non-Point Invertible Transformations of Difference and Differential-Difference Equations

Non-point invertible transformations are completely described for difference equations on the quad-graph and for their differential-difference analogues. As an illustration, these transformations are used to construct new examples of integrable equations and autotransformations of the Hietarinta equation.


Introduction
The present paper is devoted to invertible transformations for both discrete equations of the form u i+1,j+1 = F (u i,j , u i+1,j , u i,j+1 ), and "semi-discrete" chains of the differential equations (u i+1 ) x = F (x, u i , u i+1 , (u i ) x ).
Here i and j are integers, x is a continuous variable, u is a function of i, j and i, x for the first and the second equation, respectively. From now on, we shall omit i and j for brevity and, in particular, write the above equations in the form u 1,1 = F (u, u 1,0 , u 0,1 ) (1.1) and (u 1 ) x = F (x, u, u 1 , u x ). (1.2) We assume that F u F u 1,0 F u 1,0 = 0 for equation (1.1) and F ux = 0 for equation (1.2). These conditions allows us to rewrite equation (1.1) in any of the following forms Therefore, all "mixed shifts" u m,n := u i+m,j+n (for both positive and negative non-zero n and m) can be expressed in terms of dynamical variables u k,0 , u 0,l by virtue of equations (1.1), (1.3)-(1.5).
(A more detailed explanation of the dynamical variables, the notation u m,n and the recursive procedure of the mixed shift elimination can be found, for example, in [11,10].) Analogously, u (n) m := ∂ n u i+m /∂x n for any non-zero m ∈ Z and n ∈ N can be expressed in terms of x and dynamical variables u l := u i+l , u (k) := ∂ k u i /∂x k by virtue of equations (1.2), (1.6). The notation g [u] means that the function g depends on a finite number of the dynamical variables (and x if we consider equation (1.2)). The considerations in this paper are local (for example, we use the local implicit function theorem to obtain (1.3)-(1.6)) and, for simplicity, all functions are assumed to be locally analytical.
In addition to the point transformations v = g(u), some of the equations (1.1) and (1.2) admit non-point transformations v = g [u] which are invertible in the sense of [16]. For example, the differential substitutions v = u x − sin u 2 (1.7) maps solutions of the differential-difference sine-Gordon equation [7,14] (u 1 ) x − sin u 1 = u x + sin u (1. 8) into solutions of the equation which is a semi-discrete version of the complex sine-Gordon equation. Here the sign of the righthand side of equation (1.9) coincides with the sign of the cos u value 1 . Indeed, v 1 = (u x +sin u)/2 follows from equation (1.8) and, together with (1.7), gives us The inverse transformation can be found in [12]: the formula u = π 2 ± (arcsin(v 1 − v) − π 2 ) maps any real solution of equation (1.9) into a solution of equation (1.8).
This example belongs to the following class of non-point invertible transformations introduced in [17]. Let functions ϕ(x, y, z), ψ(x, y, z) satisfy the condition ϕ y ψ z − ϕ z ψ y = 0 and equation (1.2) can be written in the form (1.10) Then we rewrite (1.10) in the form of the system express u, u x in terms of v, v 1 from (1.11) and obtain (1.12) The system (1.12) is equivalent to the equation where D x denotes the total derivative with respect to x. The substitution v = ϕ(x, u, u x ) maps solutions of (1.10) into solutions of (1.13) and the transformation u = p(x, v, v 1 ) maps solutions of (1.13) back into solutions of (1.10). It is easy to see that the same scheme can be applied to the pure discrete equations of the form ϕ(u 0,1 , u 1,1 ) = ψ(u, u 1,0 ), (1.14) where ϕ(y, z) and ψ(y, z) are functionally independent. Indeed, expressing u and u 1, we obtain and rewrite (1.16) in the form of the equivalent equation Thus, the transformation v = ϕ(u, u 1,0 ) maps solutions of (1.14) into solutions of (1.17) and the inverse transformation u = p(v, v 0,1 ) maps solutions of (1.17) back into solutions of (1.14). The transformations (1.14)-(1.17) were, in fact, used in [18] without explicit formulation of the above scheme.
The invertible transformations allow us to obtain objects associated with integrability of equations (1.13), (1.17) (such as conservation laws and higher symmetries) from the corresponding objects of equations (1.10), (1.14) because we can express shifts and derivatives of u in terms of shifts and derivatives of v. Therefore, the invertible transformations may be useful for constructing new examples of integrable equations of the form (1.1), (1.2). To illustrate this, in Section 4 we construct Darboux integrable equations related via invertible transformations to difference and differential-difference analogues of the Liouville equation. In addition, an example of constructing an equation possessing the higher symmetries is contained at the end of Section 2. In this section we also demonstrate that the scheme (1.14)-(1.17) generates autotransformations of the Hietarinta equation.

Invertible transformations of discrete equations
We let T i and T j denote the operators of the forward shifts in i and j by virtue of equation (1.1). These operators are defined by the following rules: (2.1) is called invertible if any of the dynamical variables u, u k,0 , u 0,l , k, l ∈ Z, can be expressed as a function of a finite subset of the variables We exclude all mixed variables v r,s , rs = 0, from (2.2) because we consider only the cases when the transformation maps (1.1) into an equation of the form and the mixed variables can be expressed in terms of (2.2) by virtue of this equation.
It is easy to see that any shift w = v r,s maps equation (2.3) into equation (2.3) again and the composition of the shift and an invertible transformation v = f [u] is invertible too. This leads to the following

3). Then this transformation is equivalent to either a transformation of the form
or a transformation of the form Proof . The transformation is equivalent to that of the form v = h(u, u 1,0 , . . . , u k,0 , u 0,1 , . . . , u 0,l ) (2.6) because we can eliminate "negative" variables u r,0 , u 0,s , r, s < 0 from the transformation by shifts of g. We can express u as if the transformation is invertible. Differentiating equation (2.7) with respect to u k+b,0 , we for any negative a and c, and we obtain P v a,0 = P v 0,c = 0 by differentiating equation (2.7) with respect to u a,0 and u 0,c . Therefore, either The latter equality means that either T −1 i (h) =h(u, u 1,0 , . . . , u k−1,0 , u 0,1 , . . . , u 0,l ) or T −1 j (h) =h(u, u 1,0 , . . . , u k,0 , u 0,1 , . . . , u 0,l−1 ), i.e. any invertible transformation of the form (2.6) with kl = 0 is equivalent to a transformatioñ v =h(u, u 1,0 , . . . , uk ,0 , u 0,1 , . . . , u 0,l ) such thatkl < kl. Applying this conclusion several times, we obtain that (2.6) is equivalent to a transformation w = f (u, u 1,0 , . . . , u m,0 , u 0,1 , . . . , u 0,n ) with mn = 0. Proof . If f u = 0 and s is the smallest integer for which f u s,0 = 0, then the equivalent transfor- ) depends on u. Therefore, we can, without loss of generality, assume that f u = 0. We also can write because the transformation is invertible. Here the notation l = 0, m means that l runs over all integers from 0 to m. Differentiating these equalities with respect to u a,0 , we obtain Let c < 0 and s be the biggest negative integer such that (T s This implies T c j (f ) = g(u, u 1,0 , . . . , u m,0 ) and If c ≥ 0, then equations (2.8) holds too, with g = f andd = d.
It is not always easy to see whether equation (1.1) can be represented in the form (1.14). For example, at first glance it seems that the equation does not admit an invertible transformation of the form u = ϕ(v, v 1,0 ). But in reality we can rewrite this equation as and relate it to the equation Therefore, it is useful to reformulate our result in the following form.
Returning to equations (2.9), (2.10), we note that equation (2.10) was introduced in [13] in a slightly different form. This equation has also been used in [10] as an example of an equation which is inconsistent around the cube (in the sense of [1]) but possesses the higher symmetries. Therefore, we can obtain symmetries of equation ( Because L F (ξ[u]) = 0 by definition of symmetry, we see that v τ = f u 0,1 T j (ξ[u]) + f u ξ[u] (after rewriting in terms of v and its shifts) is a symmetry of equation (2.3). Applying this, for example, to the three-point symmetries The Hietarinta [6] equation 2 u 1,1 (u + β)(u 0,1 + α) = u 0,1 (u + α)(u 1,0 + β) (2.13) is another interesting example. The invertible transformations v = u 1,0 (u + α) u − α, w = βu 0,1 β + u − u 0,1 map this equation into equation (2.13) again. In addition, the Hietarinta equation is linearizable [15]. We note that the above properties of equation (2.13) are similar to those of the continuous equation which was considered in [16]. Here

Invertible transformations of dif ferential-dif ference equations
i.e. D x is the total derivative with respect to x by virtue of equations (1.2), (1.6). The inverse (backward) shift operator T −1 is defined in the similar way.
Definition 5. We say that a transformation v = f [u] maps equation (1.2) into an equation (1.2) is called invertible if any of the dynamical variables u, u k , k ∈ Z, u (l) , l ∈ N can be expressed as a function of a finite subset of the variables It is easy to see that a transformation of the form (3.2) or (3.3) is non-point only if f depends on more than one of the variables u, u 1 , . . . , u m or on at least one of the variables u (1) , . . . , u (n) , respectively.

Theorem 2. Let a non-point invertible transformation of the form (3.3) map equation (1.2) into equation (3.1). Then equation (1.2) can be written in the form
where ϕ(x, y, z) and ψ(x, y, z) satisfy the condition ϕ y ψ z − ϕ z ψ y = 0, and the transformation is equivalent to the composition of the invertible transformation w = ϕ(x, u, u x ) and an invertible transformation of the form v = h(x, w, w (1) , w (2) , . . . , w (n−1) ). In particular, any non-point invertible transformation of the form v = f (x, u, u x ) is equivalent to the composition of the transformation w = ϕ(x, u, u x ) and a point transformation v = h(x, w).  For brevity, we omit the proofs of the above propositions because they are very similar to the proofs for discrete equations.

Examples: the transformations of Liouville equation analogues
A special class of integrable equations of the form consists of equations for which there exist both a differential substitution of the form v = X(x, y, u x , u xx , . . . ) and a substitution of the form w = Y (x, y, u y , u yy , . . . ) that map (4.1) into the equations v y = 0 and w x = 0, respectively. Such equations are called Darboux integrable or equations of the Liouville type. They not only are C-integrable (in accordance with the term of [3]) but also possess infinitely many symmetries of arbitrary high order [19,20]. The complete classification of the Darboux integrable equations (4.1) has been performed in [20]. Equations with the analogous properties exist among equations of the form (1.1) and (1.2) too, but the classification of such equations is completed for a special case of equation (1.2) only [5]. Therefore, deriving new examples of discrete and semi-discrete Darboux integrable equations from already known equations may be useful (for example, to check the completeness of a future classification).

Discrete equations
The first example is the discrete Liouville equation from [9]. According to [2], this equation has the integrals maps solutions of the equation into solutions of (4.2).
Applying Corollary 1, we see that the other discrete version [8] of the Liouville equation does not admit a non-point invertible transformation. This equation is mapped into (4.2) via the non-invertible transformation u = v 1,0 v 0,1 and has the integrals

Differential-difference equations
Let us consider the following analogue of the Liouville equation: This equation has the integrals into solutions of (4.7). The above information and some other details about equation (4.7) can be found in [2]. Equation (4.7) can be written as Applying the scheme (1.10)-(1.13), we obtain v = 1 2 of equation (4.10) from the general solution z = α i + β(x) of (4.9), where α i and β(x) are arbitrary. Equation (4.10) was used in [2] as an example of an equation admitting the integrals and the solution (4.11) was constructed in this article by another method (directly form the equation X[v] = ξ(x)). Moreover, equation (4.7) can be represented in the form (1.13) too. Applying the scheme (1.10)-(1.13) in the reverse order, we get and see that the invertible transformation w = u 1 /u maps (4.7) into the equation (w 1 ) x = w x w 1 w 1 + 1 w + 1 . (4.14) The later equation has the integrals