Symmetry, Integrability and Geometry: Methods and Applications Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf–Cole Transformations ⋆

In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples.


Introduction
In [3] one has provided necessary conditions for the linearizability of a vast class of real dispersive multilinear difference equations on a quad-graph (see Fig. 1). These conditions, obtained by considering the multiscale expansion up to fifth order in the perturbation parameter, are not sufficient to fix all the free parameters of the equation and one needs to make some further ansatz or to use other techniques to fix it.
So to verify these results and provide alternative ways to prove linearizability of partial difference equations on a quad-graph we write down here a set of algorithmic conditions obtained by postulating the existence of linearizing transformations, i.e. classes of transformations which reduce a given partial difference equation on a quad-graph E = E(u n,m , u n,m+1 , u n+1,m , u n+1,m+1 ) = 0, ∂E ∂u n+i,m+j = 0, i, j = 0, 1, (1.1) for a field u n,m into a linear autonomous equation forũ n,m u n,m + aũ n+1,m + bũ n,m+1 + cũ n+1,m+1 = 0 (1.2) where a, b and c are some (n, m)-independent arbitrary nonzero constants. The choice that (1.2) be autonomous is a severe restriction but is a natural simplifying ansatz when one is dealing with autonomous equations. Moreover, as (1.1), (1.2) are taken to be autonomous equations, i.e. they have no n, m dependent coefficients, they are translationally invariant under shifts in n and m. So we can with no loss of generality choose as reference point n = 0 and m = 0. This will also be assumed to be true for all the tranformations we will study in the following. We will consider here one, two points and Hopf-Cole transformations. By a one point transformation we mean a transformatioñ between (1.2) and (1.1) characterized by a function depending just from the function u 0,0 and maybe on some constant parameters. It will be a Lie point transformation if f = f 0,0 satisfies all Lie group axioms. In the following we will only assume the differentiability of the function f up to at least second order. A natural generalization is when one considers two points transformations u 0,0 = g(u 0,0 , u 0,1 ), (1.4) characterized by a function g = g 0,0 depending on two lattice points, u 0,0 and u 0,1 . The alternative choice when g = g(u 0,0 , u 1,0 ) will be studied elsewhere. Two lattice points is the minimum number of points necessary to provide in the continuous limit the first derivative and contact symmetries have been introduced by Lie as symmetries depending on first derivatives. Often contact symmetries are also called Miura transformations [10] as R. Miura introduced them to transform the KdV into the MKdV equation and have played a very important role in the integrability of the KdV equation. Equation (1.4) contains the transformation (1.3) as a subcase but here we will assume ∂g ∂u 0,1 = 0. Under this hypothesis the conditions for point transformations are not obtained as a limiting case of the ones for contact transformations. So one and two points transformations will be treated as independent cases.
In Section 2 we discuss point transformations, present the integrability conditions which ensure that the given equation is a C-integrable equation and the differential equations which define the function f . In a similar way in Section 3 we analyze two points transformations. In Section 4 we present the conditions which ensure that the given equation is a C-integrable equation and the differential equations which define the function h for the Hopf-Cole transformation. In Section 5 we shall present a few examples while in the final section we present some conclusive remarks and open problems.

Linearization by a point transformation
In this section we discuss point transformations as, being defined by a function of just one variable, they are the simplest transformation we can propose. We will state in detail the procedure used, which will be applied later in all the other cases. This procedure follows a similar one introduced in the case of the analysis of formal symmetries for integrable quad-graph equations [7,8].
In particular we describe how we get from one side the determining equations which give the transformation and on the other side the conditions under which the equation (1.1) might be linearizable. The latter are necessary conditions which the given equation has to satisfy if a point transformation which linearizes the equation exists. If the conditions are satisfied then we can solve the partial differential equation determining the transformation and get a first approximation to the point transformation. However only if also the initial determining equation is satisfied the system is linearizable.
Assuming the existence of a point transformation (1.3) which linearizes (1.1), equation (1.2) reads f 0,0 + af 1,0 + bf 0,1 + cf 1,1 = 0. (2.1) In (2.1) and in the following equations we assume u 0j and u i0 as independent variables and consequently the variable u 1,1 appearing in the last term is not independent but it can be written in term of independent variables using the equation (1.1) [7,9]. To be able to do so, we assume that (1.1) is solvable with respect to u 1,1 where, as (1.1) depends on all lattice points we must have To solve the functional equation (2.1) we apply the Abel technique [1], i.e. we rewrite it as a differential equation. The solution of its differential consequences is a necessary condition for the functional equation to be satisfied. Let us differentiate (2.1) with respect to u 0,1 and then apply the logarithmic function. We get: whereF is given bỹ We can always introduce a differential operator A such that where φ is an arbitrary function of its argument. The most general operator of this form reads where S (i) (u 0,1 , u 0,0 , u 1,0 ), i = 1, 2 are arbitrary functions of their variables. Equation (2.5) is satisfied for any function φ if There is no further condition to fix S (2) (u 0,1 , u 0,0 , u 1,0 ). Applying the operator A onto (2.4) we get which is a differential equation for f 0,1 (u 0,1 ). In (2.6) W (x) [f ; g] is the Wronskian operator with respect to the variable x of the functions f (x) and g(x), defined as The left hand side of (2.6) depends only on u 0,1 while the right hand side depends on u 0,0 , u 1,0 and u 0,1 through the given nonlinear difference equation and the up to now arbitrary function S (2) (u 0,1 , u 0,0 , u 1,0 ). So we must have As (2.7), (2.8) must be valid for any given function S (2) (u 0,1 , u 0,0 , u 1,0 ), it follows that and When (2.9), (2.10) are satisfied also K (1)
Taking into account (3.4) it turns out that (3.2) and (3.3) give the same necessary conditions. So it is sufficient to consider one of them, say (3.2). Differentiating (3.2) once with respect to u 0,1 , we get b ∂g 0,0 ∂u 0,1 (u 0,0 , u 0,1 ) + c ∂g 1,0 ∂u 1,1 (u 1,0 , u 1,1 ) ∂F ∂u 0,1 (u 0,0 , u 0,1 , u 1,0 ) = 0. (3.5) Applying the logarithmic function to (3.5) we get the differential difference equation for the function g whereF is an explicit function given in term of the given quad-graph partial difference equation E. A solution of (3.6) could be obtained by summing it up. However in this case the resulting solution g would not be of the required form (1.4). So, to find a solution of (3.6) of the required form, we simplify (3.6) by introducing a differential operator A such that where φ is an arbitrary function of its arguments. The most general operator of this form reads a (u 0,0 , u 0,1 , u 1,0 ), i = 1, 2 are arbitrary functions of the independent variables to be determined. Equation (3.7) is satisfied for any function φ if Applying the operator A onto (3.6) and defining ψ(u 0,0 , u 0,1 ) . = log ∂g 0,0 ∂u 0,1 , we get Equation (3.10) is a differential equation for the function ψ(u 0,0 , u 0,1 ), i.e. for the function characterizing the two points transformation whose coefficients depend on the given quad-graph partial difference equation E. In (3.10) the function ψ depends just on u 0,0 , u 0,1 while the terms depending on the given quad-graph partial difference equation E depend on u 0,0 , u 0,1 , u 1,0 . As the quad-graph equation E depends also on the variable u 1,0 , (3.10) will be an equation determining the two points transformation only if some further compatibility conditions are satisfied. Differentiating (3.10) once with respect to u 1,0 , we get the following alternatives: 1. If W (u 1,0 ) [F ,u 0,1 ; F ,u 0,0 ] = 0 identically, we must have ∂ ∂u 1,0 R = 0, (3.11) which is a necessary condition for the linearizability of (1.1) through the two points transformation (1.4).

If
As the left hand side of (3.21) is independent of u 1,0 , we get the necessary condition It is straightforward to prove that, if (3.23) is satisfied, also ∂ ∂u 1,0 Q = 0 will be true. Moreover the compatibility of (3.21), (3.22) gives another necessary condition We summarize the results so far obtained in the following theorems: (a) If M = 0, apart from the linearizability conditions (3.14), (3.15), (3.23), (3.24) the following compatible conditions must be satisfied: we get the following linearizability conditions 3. F = 0, K = 0. Apart from the linearizability conditions (3.24), (3.23), (3.11) we have different results according to the value of R u 0,0 .
we get the further linearizability conditions 4. F = 0, K = 0. Apart from the linearizability conditions (3.20), (3.11) we have a set of conditions for the functions F and K involved, depending if not. These conditions are obtained by requiring that the overdetermined system obtained by explicitating (3.10), (3.19) in term of g = g 0,0 and possibly shifting be solvable for any u 0,0 , u 0,1 , u 1,0 . These equations are easy to derive by symbolic manipulation but too long to write down. So, for the sake of clarity, we do not write them down here.
Theorem 4. Given a partial difference equation (1.1) on a quad-graph, if Theorem 3 is satisfied, depending on the values of F and K, we have different partial differential equations defining the two points transformation. and ,u 0,1 .
2. F = 0, K = 0. We have different results according to the value of T u 0,0 .
3. F = 0, K = 0. We have different results according to the value of R ,u 0,0 .
(a) If R ,u 0,0 = 0 the two points transformation is obtained by solving for g = g 0,0 the compatible system of partial differential equations (3.21), (3.22) and (3.27); (b) If R ,u 0,0 = 0 the two points transformation is obtained by solving the compatible partial differential equations (3.22) and 4. F = 0, K = 0 the two points transformation is obtained by solving for g = g 0,0 the compatible system of partial differential equations (3.27) and (3.26).
When we have solved the PDEs for the function g(u 0,0 , u 0,1 ) we may still have arbitrary functions or arbitrary constants. These get fixed by inserting the function g into the equations (3.5), (3.16) and solving them and their consequences. At the end we need to verify that (3.1) or its shifted versions (3.2), (3.3) be satisfied. and in terms of h alone as

Linearization by a generalized Hopf-Cole transformation
If we assume h = h(u 0,0 ), then, differentiating (4.2) with respect to u 0,1 , we find and, by carrying out the same kind of calculations as in Section 2, we find the same linearizability conditions (2.9), (2.10) as for point transformations, i.e. Theorem 1 will be valid. However the differential equation for the transformation is different and is given by If the left hand side of (4.2) depends on u 0,0 and u 0,1 , the first left term in the right hand side depends on u 1,0 and u 1,1 and the second one on u 0,1 , u 0,2 , u 1,1 and u 1,2 . The variable u 1,1 is given in terms of the independent variables by (2.2) while u 1,2 can be rewritten in term of the independent variables as So, as from (4.5) the expression of u 1,2 in terms of the independent variables depends twice on the quad-graph equation (1.1), we will consider in place of (4.1) the equation, which in terms of h alone read The left hand side of (4.6) depends on u 0,0 and u 0,−1 , the first left term in the right hand side depends on u 1,0 and u 1,−1 = K(u 0,−1 , u 1,0 , u 0,0 ) and the second one on u 0,1 , u 0,0 , u 1,1 = F (u 0,1 , u 1,0 , u 0,0 ) and u 1,0 . So, the term on the left hand side of (4.6) depends on u 0,0 and u 0,−1 , the first left term on the right hand side depends on u 1,0 , u 0,0 and u 0,−1 while the second one on u 0,1 , u 0,0 and u 1,0 . Thus one can see that the three terms appearing in the equation (4.6) contain no overlapping set of variables. This is a condition necessary to get out of (4.6) some differential conditions for the functions F and K, i.e. for the equation (1.1) to be rewritable as the compatibility condition of (1.5) and (1.6). Let us consider (4.6) and, as we have products, we reduce it to a sum of terms by applying to it the logarithmic function. Then we differentiate the resulting equation with respect to u 0,1 . Only the second term on the r.h.s. of the equality depends on u 0,1 through the dependence of h 1,0 on u 1,1 and of h 0,0 . So we get: equivalent, in structure to (3.5). The term on the l.h.s. of (4.7) depends on u 0,0 and u 0,1 while the first factor on the r.h.s. depends on u 1,0 and u 1,1 and we can always introduce the differential operator A as given by (3.8). So, if we apply again the logarithmic function to equation (4.7) and then the operator A onto the resulting equation, setting ψ(u 0,0 , u 0,1 ) . = log ∂ ∂u 0,1 log(h + 1/b), we get the linear differential equation (3.10). It is worthwhile to notice that, even if the differential equation is the same when expressed in term of the variable ψ, its expression in term of f is different from the one in term of h.
Let us now differentiate (4.6) with respect to u 0,−1 . Proceeding in an analogous way as we did before, we get The term on the l.h.s. of (4.8) depends on u 0,−1 and u 0,0 while the first factor on the r.h.s. depends on u 1,−1 and u 1,0 . We can always introduce the differential operator B as given by (3.17). So if we apply again the logarithmic function to equation (4.8) and then the operator B onto the resulting equation, setting φ(u 0,−1 , u 0,0 ) . = log ∂ ∂u 0,−1 log h 0,−1 , we get the linear differential equation (3.19) for φ. However, as before, even if the differential equation is the same when expressed in term of the variable φ, its expression in term of f is different from the one in term of h.
As the determining equations in terms of ψ and φ are exactly the same as those of contact transformations, the linearizability conditions are as presented in Theorem 3. However the function ψ and φ are defined here differently then in the case of two points transformations. So the equations defining h = h 0,0 are different. In particular (3.26) and (3.27) in this case have to be replaced by the nonlinear equations A simpler and sometimes more useful nonlinear equation for h 0,0 can be obtained in the following way. Let us shift (4.8) by T 2 . In such a way we get Then from (4.7), (4.9) we extract the partial derivatives of h 0,0 with respect to u 0,0 and u 0,1 , Dividing (4.10) by (4.11) we get the following equation: Differentiating (4.12) with respect to u 1,0 we obtain a second order nonlinear differential equation for the function χ(u 0,0 , u 0,1 ) = log(h 0,0 ). We have: where , H(u 0,0 , u 1,0 , u 0,1 ) . = T 2 ∂K ∂u 0,−1 u 1,1 →F (u 0,0 ,u 1,0 ,u 0,1 ) .
The first and second terms of (4.13) depend on derivatives of the unknown function χ(u 0,0 , u 0,1 ) but the coefficient of the second term and the last one may contain also u −1,0 . So we have a further set of linearizability conditions. If ∂ ∂u −1,0 C = 0, differentiating (4.13) with respect to u −1,0 we have while, if ∂ ∂u −1,0 C = 0, after a differentiation with respect to u −1,0 , we have (4.14) In the first case, the solutions of (4.13) provides us with an ansatz of the function h, otherwise the function h is obtained by solving the following overdetermined system of nonlinear partial differential equations If the condition (4.14) is satisfied, then ∂ ∂u −1,0 a further linearizability condition. Equations (4.13), (4.15) are a nonlinear partial differential system which, introducing the function which is a Hopf-like equation whose solution can be obtained for example by separation of variables. Once we have a solution, we can introduce it into the lowest order differential equations and define the arbitrary functions or constant involved. The so obtained function h will provide us with a linearizing generalized Hopf-Cole transformation if the difference relation (4.2) is satisfied. Equation (4.20) can be introduced in (4.10), (4.11) and after some manipulations and the application of the operator A defined in (3.8), (3.9), we obtain a linear evolution equation for the function θ(u 0,0 , u 0,1 ) θ ,u 0,1 − F ,u 0,1 F ,u 0,0 θ ,u 0,0 =T (u 0,0 , u 1,0 , u 0,1 ) .
As in the case of contact transformations, the combination of the two cases defined by (4.18) or (4.19) and the two cases defined by (4.21) or (4.22), (4.23) gives a total of four subcases for the specification of the function θ. If the conditions (4.14), (4.16) are satisfied, the solution of the two equations (4.19) is given by where α and β are some fixed values of the variables u 0,0 and u 0,1 at which the integrals are well defined, while γ is an arbitrary integration constant.

Examples
Here we consider the linearizability conditions in the case of some interesting examples.

Liouville equation
Let us consider the discrete Liouville equation [12] u 1,1 = (u 1,0 − 1)(u 0,1 − 1) u 0,0 . = F (u 0,0 , u 0,1 , u 1,0 ). (5.1) In [12] it was shown that the transformation We can try to linearize by a two points transformation of the form (1.4). As F = K = 0 identically, we are in the fourth case. Moreover the two linearizabilty conditions (3.11), (3.20) are identically satisfied and the overdetermined system of differential equations (3.26) reads g u 0,1 = 0, whose solution is given by where C and D are arbitrary constants and θ = 0 is an arbitrary function of its arguments. As one can see, the system (5.2) does not specify the two points transformation. To define it we need to introduce (5.3) into (3.5), (3.16). In this way we get a system of two first order differential-difference equations involving θ and T 1 θ. From them we can extract a first order ordinary differential equation for θ which depends on ξ and u 1,0 . As a consequence this equation splits into an overdetermined system of two first order ordinary differential equations for θ(ξ), whose solution is given by where α is an arbitrary constant and C = 0. Hence, after a reparametrization of D, g(u 0,0 , u 0,1 ) = C[log(u 0,0 + u 0,1 − 1) + D].

(5.4)
A necessary condition for (5.4) to be a linearizing transformation, is that (3.1) be identically satisfied modulo (5.1). It is easy to show that it is not possible to find a value of D and C = 0 such that this condition is satisfied. In conclusion the equation (5.1) cannot be linearized by a two points transformation.
If we consider the linearization through a Hopf-Cole transformation, we are in the case when F = K = 0 and the linearizability conditions R ,u 1,0 = U ,u 1,0 = 0 are also satisfied. The equations for the functions ψ and φ read and their solution imply where σ, ρ, κ and τ are arbitrary nonzero functions of their argument. The function θ(u 0,0 , u 0,1 ) defined in (4.17), is specified by the conditions C ,u −1,0 = F = 0. The necessary conditions D ,u −1,0 =T ,u 1,0 = 0 are satisfied and the two equations (4.18), (4.21) respectively read The only admissible solution of this system is a constant. The solution of the overdetermined system of the two functional equations (5.5) and of the Hopf-like partial differential equation (4.20), after a reparametrization of the constant θ, is given by where γ = 0, δ andθ = 0 are arbitrary constants. A necessary condition to obtain a linearizing transformation is that (4.1) be identically satisfied modulo (5.1). No nonzero value of a, b, c, γ,θ and δ can satisfy this condition, hence (5.1) cannot be linearized by a Hopf-Cole transformation too.

Second Liouville equation
Let us consider the following version of the discrete Liouville equation As shown in [12], the noninvertible transformation u 0,0 = w 1,0 w 0,1 maps solutions of (5.6) into solutions of (5.1). Let us look for a linearizing point transformation. The necessary conditions (2.10) are identically satisfied while condition (2.9) reads −1/u 3 0,0 = 0. Hence we can conclude that (5.6) cannot be linearized by a point transformation.
If we consider the linearization through a Hopf-Cole transformation, we are in the case where F = 0, K = 0 and each of the two equations (4.7), (4.8) splits into two equations The necessary conditions (S ,u 1,0 , V ,u 1,0 ) = (0, 0) are respected and the solutions of the two equations respectively imply where α and β are two arbitrary nonzero constants and ρ and τ are arbitrary nonzero functions of their arguments. It is not difficult to see that no α, β, θ and τ exist, giving a nontrivial, e.g. nonconstant, solution for h. Hence (5.6) cannot be linearized by a Hopf-Cole transformation too.

Q + equation linearizable upto 5 th order by a multiple scale expansion [3]
Let us consider the equation ζu 0,0 u 1,0 u 0,1 u 1,1 + a 1 (u 0,0 + u 1,1 ) + a 2 (u 1,0 + u 0,1 ) + γ 1 u 0,0 u 1,1 where a 1 , a 2 , γ 1 , ζ are arbitrary real parameters with |a 1 | = |a 2 | and a j = 0, j = 1, 2. In [3] it has been shown that this equation passes a linearizability test based on multiscale analysis up to fifth order in the perturbation parameter for small u. In this case the linearizing point transformation obtained integrating (2.6) is given by Let us consider the case ζ = (a 1 + a 2 )γ 3 1 /(4a 3 1 ). If we search for a linearizing two points transformation, as a j = 0, j = 1, 2, we are always in the subcase (a). Then the necessary condition (3.14) cannot be satisfied. So, if the condition (5.8) is not satisfied, (5.7) cannot be linearized by a two points transformation.
We can try to linearize (5.7) by a Hopf-Cole transformation. We are in the case where F = 0 and the equation (4.7) splits into two equations. As the necessary condition S ,u 1,0 = 0 cannot be satisfied, (5.7) cannot be linearized by a Hopf-Cole transformation. We can conclude that, if the condition (5.8) is not satisfied, (5.7) cannot be linearized by neither a point, nor contact or Hopf-Cole transformation.

Linearizing one point transformation
The necessary conditions (2.9), (2.10) of linearizability through a point transformation are identically satisfied and the integration of equation (2.6) gives where A = 0 and B are arbitrary constants. One can easily see that no values of the constants B, a = 0, b = 0 and c = 0 exist for which the function f (u 0,0 ) satisfies (2.4) identically modulo the Hietarinta equation. As a consequence (5.10) cannot be linearized by a point transformation.

Linearizing two points transformation
As F = K = 0, we are in the case (4). Moreover the two linearizability conditions (3.11), (3.20) are identically satisfied and the overdetermined system of differential equations (3.26) reads g ,u 0,0 = 0, (5.11) The solution of (5.11) is given by where A and B are arbitrary constants and θ = 0 is an arbitrary function of its argument. As one can see, the system (5.11) is not sufficient to specify the eventual two points transformation. We need to introduce (5.12) into (3.5), (3.16). In this way we get a system of first order differential equations involving θ and T 1 θ. From them we can extract a first order ordinary differential equation for θ which depends on ξ, u 0,0 and u 0,1 . As a consequence this equation splits into an overdetermined system of four ordinary differential equations for θ(ξ). This system has no solution for generic e j , o j , j = 1, 2. As a consequence the Hietarinta equation cannot be linearized by a two points transformation.

Linearizing one point Hopf-Cole transformation
We are in the case when (2.9), (2.10) are satisfied, and the integration of (4. Through the gauge transformationũ 0,0 . = (−b/c) n (−b) −m w 0,0 we get a simplified linearizing transformation Is is moreover straightforward to demonstrate that if (5.13), (5.14) are satisfied, then also the Hietarinta equation is satisfied.

Linearizing two point Hopf-Cole transformation [11]
We are in the case defined by the conditions C ,u −1,0 = D ,u −1,0 = T ,u 1,0 = F = 0 and thus the linearizing function is defined by the equations (4.18), (4.21) which read where β, γ and δ are arbitrary integration constants. A necessary condition to obtain the linearization is that (4.1) be identically satisfied for all u 0,0 , u 1,0 , u 0,1 , u 0,2 , e j , o j , j = 1, 2 modulo the Hietarinta equation, from which we get When we insert the obtained values of e α , δ and b into the transformation (5.15), the two equations for ψ and φ are identically satisfied. As h depends on u 0,0 and u 0,1 , it is necessary that β = 0. By redefining γ . = βǫ we can eliminate the parameter β from the transformation. The transformation as well as the coefficient b of the linear equation so far obtained depend in a multiplicative way from the ratio a/c. Hence, performing the transformationũ n,m . = χ m v n,m , where χ is a constant, the linear equation (1.2) and the Hopf-Cole transformation (1.4) respectively read v n,m + av n+1,m + bχv n,m+1 + cχv n+1,m+1 = 0, v n,m+1 = 1 χ h(u n,m , u n,m+1 )v n,m , and choosing χ = −a/c we can remove the ratio a/c from the expressions of h and b. In other words we can always choose a "gauge" for the linearizing transformation in which a = −c. In conclusion we have

Solution of the Hietarinta equation
The general integral of the Hietarinta equation is obtained inserting in the inverse of the second relation (5.24) the solution of the initial-boundary value problem for the linear equation given in (5.24). This solution is given by and C represents a counterclockwise circumference in the complex z−plane, centered in z = 0 and lying inside the region of convergence of the series +∞ j=0w j,m z −j .
All these equations are linearizable through the transformation v 0,0 = arctan(u 0,0 ) and give the following linear equations With these values of the coefficients (a, b, c), (2.1) is identically satisfied modulo the corresponding equations. In the first three cases its right hand side is equal to pπ for arbitrary B while in the fourth case it is equal to B + pπ. As B is arbitrary, for the sake of simplicity we will choose B = 0 in all cases.

Conclusions
In this paper we have considered the possibility of linearizing nonlinear partial difference equations defined on a quad-graph by the use of point one, two points and a Hopf-Cole transformation. Imposing the existence of such transformations we obtained for the equations some necessary linearizability conditions and some differential equations for the transformations. We have applied our results to some nonlinear partial difference equations, some of which already known to be linearizable, to test our procedure. In the case of the nonlinear equation (5.7), obtained as a result of a classification of multilinear equations on the quad-graph by the multiple scales expansion up to fifth order, we have been able to show that all linearizability conditions considered here imply the constraint on the coefficients (5.8).
In the case of the Hietarinta equation we have been able to find out a new linearizing Hopf-Cole transformation depending just on one function, up to our knowledge unknown.
In the verification of the examples presented in [13], we discovered an accidental misprint in (5.27), as in the original article the signs of v 1,0 and v 0,1 are inverted.
A few problems are still open. We have just considered here the case of one point, two points and two points Hopf-Cole transformations but it could be interesting to consider in the future more general cases, maybe combining it with results on the integrability of the nonlinear equations. Moreover one would like to include nonautonomous transformations and lattice dependent linear equations.