Families of Integrable Equations

We present a method to obtain families of lattice equations. Specifically we focus on two of such families, which include 3-parameters and their members are connected through B\"acklund transformations. At least one of the members of each family is integrable, hence the whole family inherits some integrability properties.

some properties from the distinguished member. We stress that solutions of each member of the family can be obtained from solutions of the integrable member by discrete quadratures (which can be regarded as sort of Bäcklund transformation) and in this sense each member of the family is integrable. However, we discuss here hallmarks of integrability of the members of the family such as consistency around the cube property or τ -function formulation. Notion of the family of discrete integrable systems should not be confused with notion of hierarchies of integrable systems. The later notion was widely investigated in the literature whereas for the former one we can indicate only the articles that investigate the family of discrete KdV equations [19] and the family of discrete Boussinesq equations [20,21,22].
We discuss here two examples, the first one is continuation of our previous paper [18]. We introduce a family of difference equations associated with type III of maps discussed in [12,13] (we introduced families related to types IV and V in [18]). Example of the map of type III is a map and the three parameter family of equations (see Section 3) reads where u and v are given implicitly by a ln u + p bu + c function ψ is dependent variable on Z 2 and we denote ψ(m, n) =: ψ, ψ(m + 1, n) =: ψ 1 , ψ(m, n + 1) =: ψ 2 , ψ(m + 1, n + 1) =: ψ 12 , p := p(m) and q := q(n) are given functions of a single variable and a, b and c are arbitrary constants (we assume that one of the constants a, b or c is not equal to zero). However, ought to possible branching in formulas (1.2), the system (1.1), (1.2) needs specifying (as it was pointed us by Professors Frank Nijhoff and Yuri Suris). The specification is achieved by demanding that functions u and v obey After this specification there is still some freedom left in finding solutions of (1.1), (1.2) for given initial conditions on ψ. The freedom lies in finding the initial conditions for u and v out of initial conditions on ψ by means of (1.2). The solution need not to be unique. All the equations within the family are consistent around the cube (for the consistency around the cube property see [23,24,25,26], notice we resign from multiaffinity assumption of paper [25]). We find especially interesting the fact that we obtain examples of lattice equations together with transformations which can be regarded as Bäcklund transformations but not in the usual sense; we usually require Bäcklund transformation to be linearisable (see Definition 5) and this requirement is violated in these examples. Therefore Lax pair could not be easily found from this sort of Bäcklund transformation and it is not clear if the Lax pair exists in these cases. Members of the family are Hirota's sine-Gordon equation (choice of parameters b = 0 = c) referred also to as lattice potential modified KdV [27,28,29,30,31,14,32] (see Section 2 where we discuss various forms of lattice equations) and lattice Schwarzian KdV [28] in a disguise, see Section 2 (choice of parameters a = 0 = b or a = 0 = c) p 2 (y 12 + y 1 )(y 2 + y) = q 2 (y 12 + y 2 )(y 1 + y).
In the second example we go away from the maps of papers [12,13] and consider the map which gives also a 3 parameter family of equations (see Section 5) including Hirota's KdV lattice equation [5] x 12 − x = κ 1 and two further bilinear equations y 1 y − y 12 y 1 = κ(y 12 y + y 1 y 2 ), z 12 z + z 1 z 2 = z 12 z 2 + z 12 z 1 .
In this case an interesting fact is that the procedure yields τ -function representation of the family (see e.g. [19]) In Section 2, we give an overview of point transformations, Bäcklund transformations and difference substitutions and touch the issue of equivalence of lattice equations. We proceed in Section 3 where we present the method that leads to families of lattice equations. In Section 4 we relate our findings to some results of the papers [12,13], followed by Section 5 where we deal with Hirota's KdV lattice equation. Then we explain how to get Bäcklund transformation between members of the families (Section 6) and we end the paper with some conclusions and perspectives for future work.
2 Point transformations, dif ference substitutions, Bäcklund transformations and equivalence of lattice equations Before we start we would like to give some definitions and recall some well known relations [29,30,31,19,32] between equations that appear in the article (terminology used by various authors is far from being unified). Let us consider k dependent variables of n independent ones: u i (m 1 , . . . , m n ), i = 1, . . . , k. We denote M ≡ (m 1 , . . . , m n ).
H3 0 equation from ABS list [25] by means of point transformation x = i m+n e 2i(−1) n ψ . H3 0 in turn can be transformed into lattice potential modified KdV by substitution x = i m+n w.
Definition 4 (difference substitutions). Let j points M i , i = 1, . . . , j of a lattice are given. By difference substitution of order j we understand a transformatioñ Every point transformation is difference substitutions of order 1. Standard examples of difference substitution (of order 2, 3 and 4 respectively) are between lattice potential KdV between H3 0 (Hirota's sine-Gordon or lattice modified potential KdV) and Hirota's difference KdV; • and finally the introduction of τ function which transform every solution of the compatible system to solution of Hirota's difference KdV.
To the end we propose draft definition of Bäcklund transformation which is convenient for our purposes. However we are aware that the definition is not exhaustive (some transformation that deserve this name can be not covered by the definition).
Definition 5 (Bäcklund transformations (in narrow sense)). By Bäcklund transformation we understand here a transformation between two equations F (u 12 , u 1 , u 2 , u) = 0 and F (ũ 12 ,ũ 1 ,ũ 2 ,ũ) = 0 which is invertible to where functions f and g are fractional linear inũ and functionsf ,g are function fractional linear in u.
A classical example of Bäcklund transformation between is the transformation

Outline of the method
We consider the Z 2 lattice together with its horizontal edges (which can be viewed as set of ordered pair of points of Z 2 , i.e. E h = ((m, n), (m + 1, n))|(m, n) ∈ Z 2 ) and the vertical ones (E v = ((m, n), (m, n + 1))|(m, n) ∈ Z 2 ). We take into account a function u which is given on horizontal edges u : E h → C and a function v given on vertical ones v : E v → C. Shift operators T 1 and T 2 act on horizontal edges in standard way T 1 ((m, n), (m + 1, n)) := ((m + 1, n), (m + 2, n)), T 2 ((m, n), (m + 1, n)) := ((m, n + 1), (m + 1, n + 1)) (and similarly for vertical edges). We use convention to denote shift action on a function by subscripts T 1 u := u 1 . Now, the outline of the method we developed in [18] can be presented as follows.

From equations to involutive maps. Idea system
Take a function x given on vertices of the lattice and which obeys H3 0 equation Introduce fields u and v given on horizontal and vertical edges respectively (fields u and v are actually the invariants of a symmetry group of the lattice equation (3.1) as it was shown in [13]) We get and we arrive at the system of equations The main idea is to investigate system (3.2) rather than equation (3.1) itself. We dare to refer to the system (3.2) as to 2D Idea system III. The point is that the system (3.2) admits, as we shall see, three parameter family of potentials ψ given on vertices of the lattice. Every "potential image" of (3.2) we refer to as idolon (adopting Plato terminology of Ideas and idolons). First we apply the standard procedure for reinterpretation of equations on a lattice as a map. The reinterpretation is based on identification (see Fig. 1) We arrive at an involutive Yang-Baxter map that belongs to family of maps denoted by F III (see [12]).

Finding functions such that
The next step is to find such functions F and G such that for the map (3.4) holds. Anticipating facts, the functions will allow us to introduce a family of potentials in the next subsection. Differentiation of (3.5) with respect to u and v yields The equation above should hold for every value of U and V respectively. The equation has the form so F (U ) must satisfy (necessary but not sufficient condition) the ODE with some constants c 1 , c 2 and c 0 . Similarly we get Checking obtained by this way solutions we obtain and we find that for the map (3.4) the following equality holds

Potentials of the Idea systems. Idolons
Returning to equations on the lattice (by means of (3.3)) one can rewrite (3.6) as It means there exists function ψ such that where a, b, c and d are arbitrary constants (we assume that one of the constants a, b, c is not equal zero). The constant d can be always removed by redefinition ψ → ψ + 1 2 d and we neglect it System (3.7) and Idea system (3.2) give rise to so we get three parameter family of equations. Note that in general, (3.2) does not follows from (3.7) and (3.9) and therefore we will treat (3.2) as an additional condition that must be satisfied. As we have said in the introduction, choice of parameters b = 0 = c leads to equation H3 0 (2.1) whereas choice of parameters either a = 0 = b or a = 0 = c leads to equation A1 0 (2.2). Every such potential representation of the Idea system we refer to as idolon of the Idea system. To the end let us write another idolon. Namely, a = 0 yields the equation where u and v are solutions of the following quadratic equations p bu 2 + c = (ψ 1 + ψ)u, q bv 2 + c = (ψ 2 + ψ)v and we still assume that (3.2) holds.

Extension to multidimension, multidimensional consistency of idolons of I III
The system (3.2) can be extended to multidimension. We denote by s i (mind superscript!) function given on edges in i-th direction of the Z n lattice, by subscript we denote forward shift in indicated direction. The extension reads where p i is given function and can depend only on i-th independent variable. The crucial fact is the system is compatible Moreover, we have It means that there exists scalar function ψ such that a ln s i + p i bs i + c 1 s i = ψ i + ψ, i = 1, . . . , n.
where s i and s j are given implicitly by means of (3.13). Due to (3.11) the system (3.14) is multidimensionaly consistent (compatible) and we clarify what we mean by that in the following theorem (by i-th initial line we understand in what follows the set l i = {(m 1 , . . . , m n ) ∈ Z n | ∀ k = i : m k = 0} and by set of initial lines we mean l = l 1 ∪ · · · ∪ l n ) Theorem 1. For arbitrary initial condition on initial lines ψ(l) there exists solution (we do not exclude singularities) ψ of the multidimensional system (3.13), (3.14) that obeys (3.10).
Proof . Indeed, take arbitrary initial condition on initial lines ψ(l). Then choose a solution s i (l i ) (in general the value of s i is given on the edge between vertices that ψ and ψ i are given on) of the equation (this is a place when non-uniqueness may enter). We treat s i (l) as initial conditions for the system (3.10). Due to (3.11) the solution s i of (3.10) with initial conditions s i (l) exists (we admit singularities that come from zeroes of p j s i − p i s j ). Now, due to identity (3.12) there exists function ψ such that (3.13) holds and the value of ψ at the intersection of initial lines is equal to initial condition at the intersection of initial lines. Since s i obeys (3.10) ψ satisfies (3.14) as well. Finally ψ satisfies the assumed arbitrary initial condition since formulas (3.13) at initial lines coincides with (3.15).
We refer to the system (3.10) as to n-dimensional Idea system III and that is why we have denoted it by I III .

Maps
As we have already mentioned our inspiration was a survey on Yang-Baxter maps. Our goal now is to relate our findings to some results of the papers [12,13] and justify why it makes sense to talk about the Idea systems associated with maps of type III rather than single Idea system. The Idea systems are related by point transformation. Indeed, first we perform a cosmetic point transformation which in two-dimensional case after identification analogous to the one showed on the Fig. 1 yields F III map of paper [12] (F III ) : In fact by F III we understand equivalence class of Yang-Baxter maps (cf. [13]) the equations (4.1) and (4.2) belongs to. Now after the point transformation v i = u i (−1) m 1 +···+mn we get and its associated map Maps (4.3) and (4.4) are not Yang-Baxter maps but they are companions (if f : (u, v) → (U, V ) is involutive map then the map (u, V ) → (U, v) we refer to as companion of map f , cf. [12]) of Yang-Baxter maps H A III , H B III of paper [17]. The maps H A III , H B III can be obtained in two-dimensional case by the point transformation u 1 = x, u 2 = −y and w 1 = x, w 2 = − 1 qy respectively x 2 = y p px + qy x + y , y 1 = x q px + qy x + y and x 2 = y qxy + 1 pxy + 1 , y 1 = x pxy + 1 qxy + 1 and then by mentioned identification (see Fig. 1) Idea systems (H A III ) and (H B III ) cannot be extended to multidimension (in the sense of [25]). Finally, we list in the Table 1 basic identities of the maps that leads to existence of potentials of the Idea systems to illustrate how the basis changes when one changes a map. Table 1. Basic identities of the maps that leads to existence of potentials of the Idea system.

Type of the map Example of the map Identities
Hirota's KdV lattice equation As the second example we consider Hirota's KdV lattice equation [5] x 12 − x = κ 1 By the substitution u = x 1 x, v = x 2 x, we get we obtain an involutive mapping associated to system (5.1) Mapping (5.3) satisfies (this is the outcome of searching for such functions F and G that F (U )+ G(V ) = f (u) + g(v) as described in the previous section): hence (coming back to lattice variables (5.2)) we can introduce the potentials x, y and z Eliminating u and v from (5.1) we arrive at the following lattice equations , y 1 y − y 12 y 1 = κ(y 12 y + y 1 y 2 ), z 12 z + z 1 z 2 = z 12 z 2 + z 12 z 1 .

(5.5)
One can treat the equations as representatives of a three-parameter family of equations on φ corresponding to the choice of parameters b = 0 = c, a = 0 = b and a = 0 = c respectively. What more important is that from (5.4) we infer Compatibility condition that guarantees existence of function z reads from where we get Eliminating x, y and z from (5.4) we arrive at a compatible pair of bilinear forms of Hirota's KdV (cf. [19]) τ 112 τ − κτ 11 τ 2 = τ 12 τ 1 , τ 122 τ + κτ 22 τ 1 = τ 12 τ 2 .

Conclusions
In this paper we focused on two 3-parameter families of lattice equations. The first one (1.1), (1.2) and (1.3) is related to mappings of type III which were introduced in [12,13]. Two members (idolons) of the later are, the Hirota's sine-Gordon equation and the lattice Schwarzian KdV [28] in a disguise. Generally, all idolons are connected through Bäcklund transformations and they are 3D-consistent in the sense described in the paper. In the not-too-distant future we are going to investigate families of equations related to given integrable systems not only by discrete quadratures but also by Bäcklund transformation from the Definition 5.
The second family described by (5.6) and (5.1) is not 3D-consistent. Nevertheless, all of its idolons are connected through Bäcklund transformations, and since an idolon of this family is the Hirota's KdV equation, the whole family inherits some integrability properties e.g. τ -function formulation.
We would like to emphasize once more that the main object under consideration are Idea systems (3.2) (or its n-dimensional version (3.10)) and (5.1). The main observation is that the Idea systems admit three-dimensional vector space of scalar potentials (formulas (3.8) in case of two-dimensional Idea I III and (3.13) in the n-dimensional case, see also second and third formulas of (5.6)). In a forthcoming paper we will discuss all Idea systems that arise from equations of Adler-Bobenko-Suris list. In other words, we plan to investigate all mappings in [12,13], determine their associate Idea systems and put more light into integrability properties of the associated family of lattice equations. Also, it will be interesting to investigate the mappings that arise when one imposes periodic staircase initial data on these families of lattice equations. Another objective is to derive the discrete Painlevé equations associated with these families.
Finally, we will discuss the case of real-valued functions, which can lead to standard 3Dconsistent lattice equations. For instance for the idolon we proposed in [18] assuming f : Z 2 → R and a, b > 0 the only real solutions of the cubic equations are Then the real lattice equation (7.1), with u and v given by (7.2), is 3D-consistent. From another perspective, instead of dealing with the family of lattice equations, it seems more fundamental to define a model that consists of the Idea system and the associate potential equation (e.g. equations (3.2), (3.7) or (3.10), (3.13) for the multidimensional extension). Then the family of 3D-consistent (see Theorem 1) lattice equations follows naturally. But what more important, this is a new lattice model, defined in both vertices and edges of a 2D square lattice (nD lattice in the multidimensional extension). Such models have also been introduced in the recent work of Hietarinta and Viallet [33].