Integrable modifications of the Ito-Narita-Bogoyavlensky equation

We consider five-point differential-difference equations. Our aim is to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated to transformations of the first, second and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived one new completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.


Introduction
We consider differential-difference equations of the form: where v n = v n (t) is an unknown function of the continuous time t and discrete integer variable n, v n denotes the time derivative of v n , and f is a function of five variables. The oldest and the most famous integrable example of this class is the Ito-Narita-Bogoyavlensky (INB) equation [3,9,13]: Lately, equations of this class have been intensively studied by the generalized symmetry method, see e.g. [1,2,7,8]. However, the problem of description of all integrable equations of this class is far from complete. To find new integrable equations, we apply in this article an alternative approach using non-invertible discrete transformations.
More precisely, we use non-invertible transformations of the following special form: relating (1) and (2). They transform any solution v n of (1) into a solution u n of (2). Such a transformation is explicit in one direction and can be called the discrete substitution by analogy with differential substitutions in the continuous case, see e.g. [16,17], where such transformations were studied. Numerous examples of transformations of the form (3) can be found in [25] for the Volterra and Toda type equations and in [7,8] for the five-point equations (1). Different methods for the construction of discrete transformations (3) are presented in [6,24]. Equation (2) is one of the key equations of lists of integrable equations found in [7,8] as a result of the generalized symmetry classification of an important subclass of (1). Here we are going to enumerate all modifications of the INB equation (2), which are associated with transformations (3) of the orders k = 1, 2, 3, where k = k 1 − k 2 . We will also prove that the order of a possible transformation in this problem is restricted by the number five: k ≤ 5. This estimate is accurate in the sense that there exist transformations for all orders 1 ≤ k ≤ 5. Examples of transformations of any such an order will be given below. As far as we know, a classification problem of this kind is solved for the first time in the discrete case, as for continuous case, see e.g. [10].
A well-known example of a transformation of the form (3) is the discrete Miura transformation [20] relating the Volterra equation and its modification. There is a class of more simple linearizable transformations in the terminology of [6]. Such transformations are more simple than Miura type ones in the sense that the problem of finding v n by (3), starting from a given function u n , is more easy. We analyze transformations obtained in this paper to show that the most of them are linearizable. As a result of the classification we obtain a number of new integrable equations and transformations. Most of new equations belong to the class (1) and one of them is the discrete quad-equation. Among new transformations found here, one is of Miura type and two are linearizable in a non-standard way.
In Section 2 we discuss some theoretical aspects and prove a boundedness theorem for the orders of possible transformations. In Section 3 transformations of the orders 1 and 2 are classified together with corresponding modifications. Section 4 is devoted to transformations of the order 3. In Section 5 we discuss in detail the most interesting examples of equations and transformations obtained in the previous section. In Conclusion we briefly summarize results obtained in the paper.

Theoretical comments and results
The differential-difference equations (1) and discrete transformations (3) we consider in this paper are autonomous, i.e. do not explicitly depend on the discrete variable n. That is why, for brevity, we may write down equations and transformations at the point n = 0: where v = v 0 . Moreover, up to the shift of the discrete variable n, transformations (3) can be rewritten in the form: where u = u 0 . We also use a natural restriction: We are going to enumerate all modifications of the form (5) of the INB equation (2), which correspond to transformations (6) of the orders k = 1, 2, 3. Such modifications are integrable equations in the sense that they possess infinitely many conservation laws.
In fact, the INB equation (2) has conservation laws of an arbitrarily high order, which can easily be constructed by using the well-known Lax representation [3] or recursive operator [26]. Modifications (5) are related to the INB equation by transformations of the form (6) and, for this reason, also have conservation laws of an arbitrarily high order due to [25,Theorem 18]. That theorem is formulated for the Volterra type equations, but it can easily be reformulated for the case of five-point equations (5). Conservation laws for a modification (5) are constructed in an explicit way by using the corresponding transformation (6) and known conservation laws of (2) [25, Section 2.7]. Let us introduce the notation h j for any function for any integer j, in particular, h 0 = h.
If equation (5) is transformed into (2) by transformation (6), then the functions f, g must satisfy the determining equation: i.e. here D t is the operator of differentiation in virtue of (5).
Our main problem can formulated as follows: for any fixed k ≥ 1 we look for pairs of functions f, g satisfying (8). It is important that the functions v j , j ∈ Z, are considered in this problem as the independent variables. The functions f, g must identically satisfy (8) for all values of these variables.
Below we prove that the orders of possible transformations in this problem are restricted by the number five: k ≤ 5. This estimate is accurate in the sense that there exist transformations for all the orders 1 ≤ k ≤ 5. Examples of transformations of the orders k = 1, 2, 3 will be given in Proof . Let us rewrite condition (8) as: Differentiating (9) with respect to v k+2 and dividing the result by ∂f k ∂v k+2 , one has: We see that, if k ≥ 4, then the right hand side of this relation and therefore the function ∂ log g ∂v k do not depend on v 1 , v. Hence, the function g can be represented in the form: Let us divide (10) by g 2 : If k ≥ 6, then both sides of this relation depend on v k+1 , v k , . . . , v 4 only, therefore: where f (1) is a new function. As the left hand side of (12) does not depend on v k+1 , we apply the operator T −2 ∂ ∂v k+1 log and derive the following consequence: with a new function f (2) . Differentiating the main equation (9) for g and f with respect to v k+1 , we get: Then we differentiate the result with respect to v 1 : Here ∂ 2 log g ∂v k ∂v 1 = 0 due to (11), ∂ log g (1) ∂v 1 = 0 due to the definition of g (1) in (11), and ∂ 2 log g (2) ∂v k−1 ∂v 1 = 0 due to (14). Therefore ∂ 2 g ∂v k ∂v = 0, i.e.
but the last equality contradicts the conditions (7).

Modifications of the INB equation of the levels 1 and 2
We classify here modifications of the INB equation of the levels 1 and 2, which correspond to transformations of the orders 1 and 2. In the classification, we use differential consequences of the determining equation (8) for the functions f and g, as in the proof of Theorem 1, and we are trying all possible cases. If necessary, we can change v in an equation (5) and corresponding transformation (6) by using the point transformation: The transformation (6) and the time t in (5) can be changed by the transformatioñ leaving the INB equation (2) invariant. The classification will be carried out up to these autonomous point transformations (18) and (19). An exact statement of the result will be given later. First of all, let us enumerate all possible modifications of the first level.
In equation (E3) and in all the lists below i.e κ 3 = −1. That is why one has in (E3) two cases corresponding to the signs + and −. In equation (E5) and in all the lists below, the function φ satisfies the differential equation This function φ can be defined as: Here and below a and c are arbitrary constants. The constant c corresponds to a point symmetry.
Here and in all the lists below, a transformation with the number (TX) corresponds to the equation with the number (EX) for any X.
In equations (E8) and (E9), κ is defined by (20), i.e. in each of these equations there are two cases. In equation (E12) the function φ is defined by (21), while a and c are arbitrary constants. In the next list, corresponding transformations are presented.

Modifications of the INB equation of the level 3
Here we classify all integrable modifications of the INB equation of the third level. First we give a list of equations.

List 3. Modifications of the third level
Here κ is defined by (20), i.e. for each such an equation with a dependence on κ, there are two cases. Besides, a and c are arbitrary constants, while in equation (E21) the function φ is defined by (21). Let us list now the corresponding transformations. Equations (E16,E22) together with corresponding transformations were discussed in [7,8], see also references therein. Equation (E18) with a = c = 0 has been presented [6]. Transformations (T15)-(T17) are linearized as follows:

List 3 ′ . Transformations of the third order
where α, β, γ are some constants. For further discussion, we will need some auxiliary transformations.

The most interesting examples
Here we discuss in detail equations (E29) and (E28) and corresponding transformations (T29) and (T28).

Equation (E29)
Transformation (T29) corresponding to (E29) is quite similar to (T13) which is discussed in detail in [6, Section 3]. As it will be shown below, unlike the most of linearizable transformations presented in this paper, it is a composition of linearizable transformations in different directions.
In this case we need one auxiliary equatioṅ and two obviously linearizable transformations Transformation (T29) can be decomposed as it is shown in the following diagram: As we have mentioned in Section 2, transformation (T29) allows one to construct conservation laws for the modified equation (E29) in an explicit way, as this transformation is of the form (6). The construction of generalized symmetries is a more difficult problem. However, it is simplified in the case of linearizable transformations, as it is reduced to the use of linear transformations like (22). The construction of modified equations by using linearizable transformations is discussed in [7,Appendix A]. Using that theory, we construct here a generalized symmetry for (E29).
The decomposition shown in Diagram (26) allows one to construct the generalized symmetry for (E29) by using a known symmetry for the INB equation (2). It is even easier to use a symmetry, presented in [26], for its well-known modification (E7). At first we construct a generalized symmetry for (23), which is of the form: .
Now we can find a generalized symmetry for (E29), which reads , Equation (23)
In accordance with [6] it should be the Miura type transformation. However, a particular solution of this equation can easily be found: y * n ≡ −1. Therefore, unlike the case of general position, it can be linearized in an explicit way and can be solved by quadrature.
In fact, using this particular solution, let us change y n = y * n + 1/ỹ n in order to get a nonautonomous linear equation: Substituting the relation in terms of a new function x n , we rewrite (39) as the equation which has the obvious solutioñ As a result we have y = − x −1 x . Starting from a given function z n , we can find x n from (40) by using two discrete integrations, as: So, the Riccati equation (38) substituting the functions (49), we get a generalized symmetry for (50). Let us denote the right hand side of equation (E28) by Θ(z n+2 , z n+1 , z n , z n−1 , z n−2 ).
Then the resulting generalized symmetry is of the form: z n,m = Θ(z n+2,m , z n+1,m , z n,m , z n−1,m , z n−2,m ), i.e. it is defined by equation (E28). Due to the invariance of quad-equation (50) under the change of discrete variables n ↔ m, a generalized symmetry in the other direction m reads: z ′ n,m = Θ(z n,m+2 , z n,m+1 , z n,m , z n,m−1 , z n,m−2 ).
A generalized symmetry of (46) in the m-direction is presented in [12] and it is equivalent to symmetry (52) up to the invertible transformation (48,49). Formulae (49) define two Miura type transformations which transform (51) into (45). Both of them are equivalent to the discrete Miura transformation (T * 28) up to auto-transformations given in (30,31).

Conclusion
In this paper we have enumerated all integrable modifications of the form (1) of the INB equation (2) of the levels k = 1, 2, 3, which are related to it by non-invertible transformations of the form (3). Resulting Lists 1-3 contain 29 equations, see more precise formulations in Theorems 2-4. As far as we know, a classification problem of this kind is solved for the first time in the discrete case.
Corresponding discrete non-invertible transformations are presented in Lists 1 ′ − 3 ′ . We have analyzed those transformations to understand their nature. As a result we have shown that the most of them are linearizable.
We have also proved that the orders of possible transformations in this problem are restricted by the number five, see Theorem 1. This estimate is accurate in the sense that there exist transformations for each order k: 1 ≤ k ≤ 5. Transformations of the orders k = 1, 2, 3 have been completely enumerated in Sections 3,4. Examples of the orders k = 4, 5 are presented in Section 5. The complete classification of transformations of the fourth and fifth orders is left for a future work.
As a result of the classification we obtain a number of new integrable equations and noninvertible discrete transformations. Among new transformations we would like to highlight transformation (T * 28) of Miura type and two transformations (T29) and (36.a) that are linearizable in a non-standard way. Among equations we note two five-point differential-difference equations (E28), (E29) and one quad-equation (50) compatible with (E28).