## Abstract

Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the ‘space–time’ lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PΔEs possessing this property, leading to the so-called Adler–Bobenko–Suris (ABS) list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly non-trivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding PΔE the Lagrange forms are closed, i.e. they obey a *closure relation*. Here, we extend those results to the continuous case: it is known that associated with the integrable PΔEs there exist systems of partial differential equations (PDEs), in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper, we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property.

## 1. Introduction

The study of integrable systems on the space–time lattice has, over the past decades, developed into a major area of research. The early examples of integrable lattice systems, i.e. partial difference equations (PΔEs) on two- or higher-dimensional grids, go back to the mid-1970s and early 1980s, when the research was focused on discretizing known continuous soliton systems (Ablowitz & Ladik 1976, 1977; Hirota 1977; Date *et al.* 1982, 1983; Nijhoff *et al.* 1983; Quispel *et al.* 1984). However, in recent years the focus has shifted to studying these equations on their own merit, since they exhibit the key integrability aspects in a more lucid way than their continuous counterparts, and often can be thought of as more generic systems than the corresponding differential equations which can be retrieved by appropriate continuum limits^{1} .

It has emerged that one of the key integrability aspects of lattice equations is the property of multidimensional consistency, i.e. the property that one can impose PΔEs of the same form (differing only by a choice of the so-called lattice parameters) in different pairs of perpendicular directions of a multidimensional lattice in which the two-dimensional grids are embedded (Nijhoff & Walker 2001; Bobenko & Suris 2002). This phenomenon can be understood as the analogue of the existence of infinite hierarchies of PDEs associated with an integrable nonlinear evolution equation, such as the Korteweg–de Vries (KdV) equation. In some sense multidimensional consistency implies that the ‘integrable system’ is not really a single PΔE but comprises all members of the parameter-family of equations compatible on the multidimensional lattice (Nijhoff *et al.* 2001). Using this property a classification of two-dimensional scalar integrable lattice systems was given, in the affine-linear case, by Adler *et al.* (2003) resulting in what is hereafter referred to as the ABS list. Although we shall refer to the equations in the list as the ‘ABS equations’, which are given in the Appendix, several examples in the list, notably lattice systems of KdV-type (associated with discretizations of the KdV and related PDEs), were already known in the literature, cf. Nijhoff & Capel (1995) and references therein. The most general equation of the ABS list was found by Adler (1998) and it contains lattice parameters which are points on an elliptic curve, cf. also Adler & Suris (2004).

Considering the integrable system as a system of many equations imposed simultaneously on one and the same (possibly scalar) dependent variable, raises the question whether there exists a canonical description which encodes the whole system of equations. Even though most continuous integrable equations can be cast into the Lagrangian form (Zakharov & Mikhailov 1980), the situation for discrete equations is distinctly more complicated and, in fact, the first Lagrangian for a quadrilateral equation was given in Capel *et al.* (1991). Furthermore, the usual least-action principle only provides one variational equation per component of the dependent variable. Lobb & Nijhoff (2009) provided the first proposal on how to achieve a variational description of the multidimensionally consistent system of lattice equations, based on the idea of Lagrangian multi-forms. In fact, the proposal is to consider action functionals in terms of discrete Lagrange 2-forms given by
1.1
where *u*(** n**) is a field configuration of a dependent variable

*u*depending on lattice sites labelled by of a lattice of arbitrary dimension

*N*>2 and

*σ*denotes a quadrilateral surface, composed of a connected configuration of quadrilaterals, embedded in the

*N*-dimensional grid. The least-action principle formulated in Lobb & Nijhoff (2009) imposes that

*S*attains a critical point not only with regard to the infinitesimal variations of the field variables

*u*(

**) for a given chosen surface**

*n**σ*, but also with regard to the geometry of the independent variables, by imposing surface independence of the action for the solutions of the Euler–Lagrange (EL) equations. The latter requirement is equivalent to stating that the Lagrangians , viewed as 2-forms associated with the quadrilaterals that compose the surface, have to be closed forms subject to the EL equations. It is this property that makes the choice of Lagrangians admissible. Remarkably, this closure property could be established for well-chosen Lagrangians for equations in the ABS list, either by direct computation (as was done in Lobb & Nijhoff 2009) or by general principles (Bobenko & Suris 2010). Furthermore, Lagrangians possessing the closure property were found in the case of multi-component systems (Lobb & Nijhoff 2010) and higher-dimensional systems of Kadomtsev–Petviashvili (KP)-type (Lobb

*et al.*2009).

The phenomenon of multidimensional consistency is not restricted to the discrete case of lattice equations. In Nijhoff *et al.* (2000) the novel class of non-autonomous PDEs was introduced which are the equations of the same dependent variable appearing in the lattice equations but viewed now as a function of a pair of (continuous) lattice parameters, denoted by *α*_{i} and *α*_{j}. The so-called *generating PDE* of the KdV hierarchy, i.e. a PDE which generates all the equations in the KdV hierarchy by systematic expansion with respect to the lattice parameters, arises as the EL equations from a simple Lagrangian given by
1.2
In Lobb & Nijhoff (2009) it was also shown that this continuous Lagrangian can be viewed as a closed 2-form, on solutions of the EL equations, subject to a three-dimensional constraint on the solutions^{2} . Thus, a similar multidimensional least-action principle in terms of a continuous action functional
1.3
which involves continuous Lagrange 2-forms depending on the fields *u*(** α**) and on surfaces

*σ*with coordinates given by the vector

**=(**

*α**α*

_{1},…,

*α*

_{N}), where

*i*,

*j*=1,…,

*N*, can be formulated in the continuous case for certain PDEs associated with KdV-type lattice equations. In order to avoid the difficulty stemming from the fact that the scalar Lagrangian (1.2), and consequently also the EL equation, is of higher order it is natural to formulate the Lagrange structure to that for a system of lower-order equations, which actually emerge naturally from the corresponding Lax pair. In fact, such systems of PDEs were subsequently established for all equations in the ABS list through symmetry analysis (Tsoubelis & Xenitidis 2009) and as a specific example of the construction we can mention the following coupled system of PDEs, corresponding to H1, H2 and Q1: 1.4 This system was derived in Tsoubelis & Xenitidis (2009) along with its Lax pair, where

*u*,

*u*

_{i}and

*u*

_{j}denote a triple of dependent variables. It is worth mentioning that system (1.4) with

*δ*=0 was first given in Nijhoff

*et al.*(2000) in relation to H1 and was derived from a reduction of the anti-self dual Yang–Mills equations in Tongas

*et al.*(2001). In the present paper we will derive the Lagrange structure for the general class of systems of the form such that the EL equations, when varying with respect to all three dependent variables, are satisfied on solutions of the systems like equation (1.4). In this case, the Lagrangian reads 1.5 and one can show directly, without assuming further constraints on the higher-dimensional embedding, that this Lagrangian obeys the continuous version of the closure relation 1.6 The organization of the paper is as follows. We first give in §2 a universal description of the Lagrange structure of affine-linear quadrilateral equations. What is important here is that the Lagrangians can actually be systematically derived from some basic assumptions, such as a three-point form of the Lagrange function and the Kleinian symmetry of the quadrilateral equation. This leads to an integral form of the so-called three-leg representation of the quadrilateral equation, cf. Adler

*et al.*(2003). In §3, we specialize to the ABS case, in which we have the symmetries of the square, imposing dependence on lattice parameters in a covariant way. We prove that the corresponding Lagrangians, in the integral form, obey the closure property. In §4, we briefly present the construction of the universal system of PDEs associated with the ABS list, which was given in Tsoubelis & Xenitidis (2009), and we prove the multidimensional consistency of these systems. Finally, in §5, we establish the corresponding Lagrangian structures, both for the systems of PDEs as well as for the differential–difference equations which are defined in terms of the symmetry generators, and show that the universal Lagrange structure for the system of PDEs obey the closure relation.

## 2. Lagrangian formulation of two-dimensional quad-lattices

We first focus on the Lagrangian formulation of affine-linear quadrilateral equations on the two-dimensional lattice, and show that under very general assumptions there is an almost unique Lagrangian description. The members of the class of equations we consider will be denoted by
2.1
and involve the values of a function *u* at four neighbouring points of an elementary quadrilateral. The labels *i* and *j* refer, for the time being, to two fixed directions of a lattice.

The general notation we use here, adapted to cover also the situation we will encounter in the next section, is as follows. In principle, the unknown function *u* of the equations will, in due course, be considered as functions of an arbitrary number of discrete variables, denoted by *n*_{i}, and there will be an equal number of continuous variables *α*_{i}, i.e. *u*=*u*(*n*_{1},*n*_{2},…;*α*_{1},*α*_{2},…) will be the dependent variable of a system of equations in an arbitrary number of dimensions. In that context, the shifted values of *u* will be denoted by indices, e.g.
where *T*_{i} is the shift operator in the *i* direction. Occasionally, we will also use the difference operator Δ_{i}, which is defined by Δ_{i}:=*T*_{i}−1. The main integrability property of the systems considered in the following sections is multidimensional consistency. This means that all of them can be extended in a compatible way in higher dimensions, according to the following definition (Nijhoff & Walker 2001; Bobenko & Suris 2002):

### Definition 2.1

Let *Q*(*u*,*u*_{i},*u*_{j},*u*_{ij})=0 denote a quadrilateral difference equation, to be solved for a dependent variable *u*. Imposing the dependence of *u* on a third discrete variable *n*_{k} such that
if the three different ways to evaluate the triply shifted value *u*_{ijk} lead to the same result, then we call the equation defined by *Q* multidimensionally consistent.

However, in the present section we will only make statements about a single equation of the form (2.1), in terms of two fixed independent variables *n*_{i} and *n*_{j}, and their corresponding lattice shifts. The following properties of the function *Q* are assumed here:

—

*Q*is an affine-linear polynomial depending explicitly on all the arguments, that is ∂_{u}*Q*∂_{ui}*Q*∂_{uj}*Q*∂_{uij}*Q*≠0 and .—

*Q*is irreducible, meaning that*Q*cannot be factorized and presented as a product of two polynomials.—

*Q*possesses the Kleinian symmetry, i.e. 2.2 where*ϵ*=±1 and*σ*=±1.

If *Q* satisfies property 1, then one can define, in principle, six different bi-quadratic polynomials in terms of *Q* and its derivatives, and four quartic polynomials through the application of the following double-sided Wronskian, respectively, discriminant operator, defined by their actions on an arbitrary polynomial function *f*=*f*(*x*,*y*) in the following ways, e.g. Adler *et al.* (2009):
2.3

Furthermore, the irreducibility of *Q* guarantees that none of the polynomials obtained from *Q* by acting on it by the operator for all pairs of arguments *x* and *y* of *Q*, is identically zero. The Kleinian symmetry (2.2) implies that these bi-quadratic polynomials are symmetric and take the same form when we apply the operator for mutually exclusive pairs of arguments. This is expressed by the following lemma, which can be easily established by exploiting the definitions and symmetry relations, see Adler *et al.* (2003, 2009):

### Lemma 2.2

*Due to the affine linearity and the Kleinian symmetry (2.2), the following polynomials assigned to an elementary quadrilateral are the same for functions depending on complementary pairs of variables*:

*symmetric bi-quadratic polynomials assigned to the edges of the quadrilateral*: 2.4*symmetric bi-quadratic polynomials assigned to the diagonals of the quadrilateral*: 2.5*quartic polynomials associated with the vertices of the quadrilateral*: 2.6

*Furthermore, between the bi-quadratic polynomials the following relations hold on solutions of equation* (2.1):
2.7

With the above definitions we are now in a position to write an integral form of the three-leg formula for an affine-linear quadrilateral equations of the type (2.1) subject to the properties (1)–(3) given above. The usual three-leg formula was given in Adler *et al.* (2003) on a case-by-case basis. In the monograph by Bobenko & Suris (2008 p. 281), a result by Adler is cited as an exercise, of which the following is a slight generalization:

### Proposition 2.3

*Let Q possess the properties given above. Then the equation*
2.8
*holds on solutions of Q*(*u*,*u*_{i},*u*_{j},*u*_{ij})=0, *where x and y are arbitrary and where the function z is defined through the relation Q*(*u*,*x*,*y*,*z*)=0.

### Proof.

The proof is straightforward using implicit differentiation with respect to the variables *x*, *y*, *u*_{i} and *u*_{j}. Substituting *u*_{ij}=*z*(*u*,*u*_{i},*u*_{j}), since we require relation (2.8) to hold on solutions of *Q*(*u*,*u*_{i},*u*_{j},*u*_{ij})=0, and denoting the resulting left-hand side of equation (2.8) by *W*(*u*;*x*,*y*;*u*_{i},*u*_{j}), we can easily prove that *W* is actually independent of *x* and *y*, and also of *u*_{i} and *u*_{j}. To show this it is sufficient to show that ∂_{x}*W*=0 through implicit differentiation. Computing the derivative of *W* with respect to *x* we obtain:
2.9
since *z* is determined by *u*, *x* and *y* through the equation *Q*(*u*,*x*,*y*,*z*)=0. Implicit differentiation of the latter yields
2.10
which holds on solutions of *Q*(*u*,*x*,*y*,*z*)=0 by taking into account the definitions
The combination of equations (2.9) and (2.10) leads to the result ∂_{x}*W*=0. In precisely the same fashion it can be established that ∂_{y}*W*=∂_{ui}*W*=∂_{uj}*W*=0, where in the latter two we make use of the substitution for *u*_{ij} as mentioned above. Since as a consequence *W*(*u*;*x*,*y*;*u*_{i},*u*_{j}) does not depend on *x*,*y*, we can choose the latter arbitrarily, leading to the conclusion that *W*(*u*;*x*,*y*;*u*_{i},*u*_{j})=*W*(*u*;*u*_{i},*u*_{j};*u*_{i},*u*_{j}), where the latter expression vanishes due to the integration limits and taking into account that for *x*=*u*_{i}, *y*=*u*_{j} we have that *z*=*u*_{ij}. ■

Lagrangians for the generalized three-leg formula (2.8) can be obtained by simple integration over the remaining variable *u*, but there is a stronger result that holds. In fact, it turns out that under the assumption of the three-point form of the Lagrange function, as was done in Lobb & Nijhoff (2009), one can actually derive in an almost unique way the form of the Lagrange function for the affine-linear quadrilateral equations. This is expressed as follows:

### Theorem 2.4

*Let Q(u,u*_{i}*,u*_{j}*,u*_{ij}*) be an irreducible, affine linear and Kleinian-symmetric polynomial and the three-point function* *of the lattice be given by
*
2.11
*in which p and q are solutions of the equations Q(v,s,p,v*_{ij}*)=0 and Q(v,q,t,v*_{ij}*)=0, respectively. Then* *obeys the following variational equations:
*
2.12
*and
*
2.13
*on solutions of equations Q(u,u*_{i}*,u*_{j}*,u*_{ij}*)=0 and Q(v,v*_{i}*,v*_{j}*,v*_{ij}*)=0, respectively.*

*Thus, up to the freedom in equation (2.11) of adding a constant term and multiplying by a constant factor* *can be considered to be the Lagrangian for the quadrilateral equations in both the variables u and v.*

### Proof.

Let us suppose that we are looking for a Lagrangian such that the EL equation (2.12) holds on solutions of the equation *Q*=0.

First we differentiate equation (2.12) with respect to *u*_{−i} by taking into account that *u*_{−i,j} is determined by the equation *Q*(*u*_{−i},*u*,*u*_{−i,j},*u*_{j})=0. This yields
2.14
where
on solutions of equation *Q*(*u*_{−i},*u*,*u*_{−i,j},*u*_{j})=0. Thus, substituting the latter into equation (2.14) and shifting forward in the *i* direction the resulting relation, one arrives at
2.15
In the same fashion, the derivative of equation (2.12) with respect to *u*_{−j} leads to
2.16
A third equation follows from the compatibility of equations (2.15) and (2.16), namely
2.17

The above three equations can be considered as first-order PDEs for , and , respectively. Solving these equations by the method of characteristics, one arrives at the following expressions for the first-order derivatives of
2.18
where *Φ*, *Ψ* and *X* are arbitrary functions of their arguments and *v* and its shifts are also arbitrary.

The compatibility conditions among the above relations lead to ∂_{1}*Φ*=∂_{1}*Ψ*=∂_{1}*X*, where ∂_{1} denotes the derivative with respect to the first argument of the corresponding function. Since the first arguments of these functions are functionally independent^{3} , it follows that the arbitrary functions *Φ*, *Ψ* and *X* must be linear with respect to their first arguments and in particular
2.19
where *c* is constant and, for convenience, we have introduced primes which denote differentiation with respect to the argument of the corresponding functions.

The integration of the above system leads to 2.20

Now, we write out explicitly the EL equations (2.12) using equation (2.20) and taking into account the three-leg form (2.8). This leads to
where *p* and *q* are solutions of *Q*(*v*,*u*_{i},*p*,*v*_{ij})=0 and *Q*(*v*,*q*,*u*_{j},*v*_{ij})=0, respectively.

One solution of the latter equation follows by choosing
where *μ* is a constant, which in turn implies that
and
which are determined up to an additive constant.

Substituting back into equation (2.20), it is not difficult to see that the terms involving the function *χ* can be written as −*μ*Δ_{i}*χ*(*u*)−(*μ*−1)Δ_{j}*χ*(*u*). These terms do not contribute to the EL equations and can be omitted from . Hence, up to a multiplicative constant and an additive constant, is given by equation (2.11).

It is a straightforward calculation to show that Lagrangian (2.11) is actually independent of *v*. On the other hand, considering as a function of *v*_{i}, *v*_{j} and *v*_{ij}, the EL equation (2.13) when it is written out explicitly reads as follows:
Obviously, the above equation is satisfied if we choose the integration limits in each integral to coincide, i.e.
But both of the latter equations imply that *Q*(*v*,*v*_{i},*v*_{j},*v*_{ij})=0. Thus, EL equation (2.13) is satisfied on solutions of *Q*(*v*,*v*_{i},*v*_{j},*v*_{ij})=0. ■

Starting with the Lagrangian (2.11), one can derive three different Lagrangians by applying the interchanges implied by the Klein symmetry. To make this statement clear, we apply changes (2.2) to Lagrangian (2.11). The first interchange, i.e. , which leaves invariant the equation *Q*=0, when applied to leads to , which is another Lagrangian of equation *Q*=0. Additionally, applying the changes one arrives at another Lagrangian, namely . Finally, combining these two symmetries one can derive a third Lagrangian which has the form .

In Bobenko & Suris (2010) the proof of the closure relation makes use of a fundamental property of the Lagrangian, which in the context of the present approach can be generalized and understood as the following:

### Proposition 2.5

*For the Langrangian (2.11) considered as a function of u*, *respectively v*, *the relation*
2.21
*holds on solutions of Q*(*u*,*u*_{i},*u*_{j},*u*_{ij})=0, *respectively*, *Q*(*v*,*v*_{i},*v*_{j},*v*_{ij})=0.

### Proof.

To prove that the sum is constant on solutions of *Q*=0, it is sufficient to prove that its derivatives with respect to *u* and its shifts vanish on solutions of *Q*=0. For instance, writing out explicitly the derivative of equation (2.21) with respect to *u*, one arrives at the three-leg form of *Q*=0 centred at *u*, i.e. relation (2.8), which holds on solutions of *Q*=0. Similarly, the derivatives with respect to the shifted values of *u* lead to the three-leg forms of *Q*=0 centred at the corresponding value of *u*. In the same fashion, considering *u* and its shifts as parameters in equation (2.21) and *v* as solution of *Q*(*v*,*v*_{i},*v*_{j},*v*_{ij})=0, the derivatives of the left-hand side of equation (2.21) with respect to *v* and its shifts vanish by taking into account that *Q*(*v*,*v*_{i},*v*_{j},*v*_{ij})=0. ■

## 3. Lagrangian multi-forms of the ABS equations

Among the members of the class of equations studied in the previous section are the ones possessing the symmetries of the square, or D4 symmetry. These equations may depend on two lattice parameters, denoted by *α*_{i} and *α*_{j}, respectively, and their defining polynomial will be denoted by *Q*_{ij}(*u*,*u*_{i},*u*_{j},*u*_{ij}), or simply by *Q*_{ij}, in order to make this dependence explicit.

Since the lattice parameters are assigned to the edges of a plaquette, the symmetries of the square imply that the interchange of *u*_{i} and *u*_{j} must be followed by a mutual interchange of the lattice parameters, i.e.
3.1

The ABS equations, which are given in Appendix A, belong to this subclass and depend explicitly on the lattice parameters. The system of equations *Q*_{ij}=0, *Q*_{jℓ}=0, *Q*_{ℓi}=0 is multidimensionally consistent in the sense of Nijhoff & Walker (2001) and Bobenko & Suris (2002), namely for the case where the quadrilateral function *Q* depends on lattice parameters (indicated here by the indices). These two characteristics of the ABS equations have the following consequences. First of all, the polynomials *h* assigned to the edges can be factorized as
3.2
The function *κ*_{ij}=*κ*(*α*_{i},*α*_{j}) is antisymmetric, i.e. *κ*_{ji}=−*κ*_{ij}, and *h*_{i}(*x*,*y*) is a symmetric bi-quadratic polynomial of *x* and *y* and the index *i* denotes that it depends only on the lattice parameter *α*_{i}.

Moreover, the polynomial *H*(*x*,*y*) is symmetric and bi-quadratic with respect to *x* and *y*, and symmetric with respect to the lattice parameters. It is more convenient for the analysis of the ABS equations in the following sections to introduce the function
3.3
which is antisymmetric with respect to the interchange of indices : *H*_{ij}=−*H*_{ji}. Again, the indices denote the dependence on the lattice parameters and not shifts in the corresponding lattice directions. Functions *h*_{i}, *H* and *κ*_{ij} can also be found in appendix A.

Finally, the factorization of polynomials *h* implies that the three-leg form for these equations can be written as
3.4
where *x* and *y* are arbitrary and *z* is any solution of *Q*_{ij}(*u*,*x*,*y*,*z*)=0.

From the multidimensional consistency of the ABS equations one can derive an affine linear equation relating *u* and its double shifts. In particular, the equation
3.5
relating the four indicated values of *u* follows from equations *Q*_{ij}=0, *Q*_{jℓ}=0, *Q*_{ℓi}=0 by eliminating the single shifted values of *u*. Indeed, one may solve the last two equations with respect to *u*_{j} and *u*_{i}, respectively, and substitute the results into the first equation. This derivation leads to
It can be checked for all of the ABS equations that the left-hand side of the above equation is factorized as , which implies equation (3.5). Alternative derivations lead to similar expressions which follow from the above by cycling permuting indices *i*, *j* and ℓ.

Moreover, it can be checked for all the ABS equations that the following relations hold on solutions of equation (3.5).
3.6
These imply that equation (3.5) may be regarded as being defined on an elementary quadrilateral with polynomials *H*_{ij}, *H*_{jℓ} and *H*_{ℓi} assigned to its edges and diagonals. Additionally, they imply that polynomial *P* possesses the Kleinian symmetry and consequently the results of the previous section can be applied to equation *P*=0 as well. Apart from the three-leg forms given in proposition 2.3 and the Lagrangian given in theorem 2.4, there are alternative formulas which follow from the derivation of equation (3.5). For instance, adding the three-leg forms centred at *u* of the equations *Q*_{ij}=*Q*_{jℓ}=*Q*_{ℓi}=0, one arrives at an equivalent three-leg form of equation (3.5), namely
3.7
where *Q*_{ij}(*u*,*v*_{i},*v*_{j},*r*_{1})=0, *Q*_{jℓ}(*u*,*v*_{j},*v*_{ℓ},*r*_{2})=0 and *Q*_{ℓi}(*u*,*v*_{ℓ},*v*_{i},*r*_{3})=0 determine the lower limits with *v*_{i}, *v*_{j} and *v*_{ℓ} being arbitrary. Three-leg forms of equation (3.5) centred at the double shifted values of *u* can be derived in the same fashion.

It is straightforward to show that the function
3.8
is a four-point Lagrangian for equation (3.5), where the upper limits are determined by *Q*_{ij}(*s*,*v*_{i},*v*_{j},*p*_{ij})=0, *Q*_{ij}(*q*_{ij},*v*_{i},*v*_{j},*t*)=0 and similar ones by cycling permuting the indices. Additionally, three more four-point Lagrangians follow by applying the interchanges of the Kleinian symmetry (2.2) in Lagrangian (3.8), namely
and
Moreover, for the function given in equation (3.8), the following relation holds on solutions of equation (3.5).
3.9
Indeed, the derivatives of this sum with respect to *u* or any of its double shifted values vanish by taking into account equations *Q*_{ij}=0, *Q*_{jℓ}=0 and *Q*_{ℓi}=0 and their corresponding three-leg forms. Finally, the independence on the lattice parameters follows by differentiation with respect to the parameters and implementation of the following:

### Lemma 3.1

*Let h*_{i}(*u*,*x*) *be any of the polynomials related to the ABS equation Q*_{ij}=0. *Then, the relation*
3.10
*holds identically*.

*Similarly, if H*_{ij}(*u*,*x*) *is any of the diagonal polynomials corresponding to the ABS equation Q*_{ij}=0, *then the following identities hold*:
3.11

The identity for *h*_{i} follows from the symmetry analysis of the ABS equations, (Xenitidis 2009; Mikhailov *et al.* 2010), and its derivation is given in appendix B. Actually, the same relation can be found in the context of integrable chains of differential–difference equations and their master symmetries (Yamilov 2006). The relation for *H*_{ij} can be proven by direct calculations, or, in the case of Q1–Q4, using the observation that *H*_{ij}(*u*,*x*)≡−*h*(*u*,*x*,*α*_{i}−*α*_{j}).

Having introduced through theorem 2.4 a Lagrangian for the ABS equations, we now proceed to prove that it obeys the closure property.

### Theorem 3.2

*The following function
*
3.12
*constitutes a double-sided Lagrangian for the ABS equation Q*_{ij}*=0, in the sense that the EL equation (2.12) is satisfied whenever Q*_{ij}*(u,u*_{i}*,u*_{j}*,u*_{ij}*)=0 and equation (*2.13*) is satisfied whenever Q*_{ij}*(v,v*_{i}*,v*_{j}*,v*_{ij}*)=0.*

*Moreover, by choosing v (resp. u) in equation (3.12) as a solution of the ABS equation Q*_{ij}*=0, the Lagrangians* *and* *satisfy the closure relation
*
3.13
*which holds on solutions of the ABS equation Q*_{ij}*=0, Q*_{jℓ}*=0 and Q*_{ℓi}*=0 for u (resp. v), and hence, in view of skew symmetry of the Lagrangian functions* *, it defines a closed discrete 2-form.*

### Proof.

Since the ABS equations belong to a subclass of the family of equations studied in §2, the Lagrangian (3.12) easily follows from equation (2.11) by choosing the multiplicative constant to be *κ*_{ij} and using relations (3.2) and (3.3).

Writing it out explicitly the left-hand side of equation (3.13), we observe that the terms involving the integrals of polynomials *h* with respect to *u* and its single shifts cancel out. Next we use proposition 2.5 for the three Lagrangians involved in equation (3.13) to replace the integrals of polynomials *h* with respect to single- and double-shifted values of *u*. To this end, relation (3.13) becomes equation (3.9), which in turn implies that equation (3.13) holds on solutions of the corresponding ABS equations. ■

### Remark 3.3

There exist deformations of the H equations in the ABS list, which were first presented in Adler *et al.* (2009), and further studied in Xenitidis & Papageorgiou (2009). The defining functions of these equations possess the basic properties of affine linearity and irreducibility, and the symmetries of the rhombus. They are defined on a black–white lattice which implies that they can be written in non-autonomous form. In this formulation, the polynomials assigned to the edges have the same form and can be factorized like the polynomials (3.2) corresponding to the ABS equations. On the other hand, there are two diagonal polynomials *H*_{1} and *H*_{2} related to each other by shifts, e.g. *T*_{i}*H*_{1}(*u*_{i},*u*_{j})= *H*_{2}(*u*_{ii},*u*_{ij}), cf. Xenitidis & Papageorgiou (2009). Actually, from theorem 3.2 follows a Lagrangian for the deformed H equations which satisfies the closure relation as well. One has to use the polynomials *h* corresponding to the deformed equations and replace *H*_{ij}(*u*_{i},*u*_{j}) and *H*_{ij}(*u*,*u*_{ij}) by *H*_{1}_{ij}(*u*_{i},*u*_{j}) and *H*_{2}_{ij}(*u*,*u*_{ij}), respectively, in equation (3.12).

### Remark 3.4

The lift of the ABS equation *Q*_{ij}=0 to a two-field discrete system was suggested by Papageorgiou & Tongas (2009) in the construction of Yang–Baxter maps. Denoting by *u* and *v* the fields involved in this system, the latter has the following form
3.14
A Lagrangian for the above system follows actually from equation (3.12) and it has the following form:
where *u*^{0}, *v*^{0} and their shifts are arbitrary while *p* and *q* are determined by solutions of the equations and , respectively. It is a straightforward calculation to prove that the variational derivatives of the action with respect to *u* and *v* vanish on solutions of *Q*_{ij}(*v*_{−j},*v*_{i,−j},*u*,*v*_{i})=0 and *Q*_{ij}(*u*_{−i},*v*,*u*_{−i,j},*u*_{j})=0, respectively.

## 4. Multidimensionally consistent continuous systems from symmetry reductions

In this section, we summarize the derivation of continuous systems from the ABS equations and their symmetries, which was given in Tsoubelis & Xenitidis (2009), and prove that these systems are multidimensionally consistent as their discrete counterparts.

All of the ABS equations admit a pair of extended generalized symmetries which play the role of master symmetries for the corresponding generalized symmetries (Rasin & Hydon 2007; Xenitidis 2009). This pair of extended symmetries will be denoted by *V*_{i} and *V*_{j} and can be given as the differential operators
4.1

Since the equations under consideration depend explicitly on the lattice parameters, the same holds for their solutions. Thus, it is interesting to consider solutions of the ABS equations remaining invariant under the action of the extended symmetries. Such solutions satisfy the discrete equation under consideration and the system of differential–difference equations
4.2
and
4.3
From this overdetermined system Tsoubelis & Xenitidis (2009) derived the following *system of partial differential equations*:
4.4
4.5
and
4.6
where the arguments of *h*_{i}(*u*,*u*_{i}), *h*_{j}(*u*,*u*_{j}) and *H*_{ij}(*u*_{i},*u*_{j}) have been omitted and *n*_{i} and *n*_{j} were scaled by 2. Moreover,
4.7
and the coefficients in the last equation of the system are given by the following relations:
4.8
4.9
and
4.10
In what follows, system (4.5)–(4.10) will be denoted as *Σ*_{ij}.

### Remark 4.1

The derivation of *Σ*_{ij} assumes that *u*_{i} and *u*_{j} are related to *u* by shifts. Actually, this relation can be forgotten and all the functions involved in *Σ*_{ij} can be treated as independent, which is the point of view we adopt from now on: we consider *Σ*_{ij} as a closed-form coupled system for the three functions *u*, *u*_{i} and *u*_{j}.

### Remark 4.2

The two equivalent forms at the definitions (4.8)–(4.10) of the coefficients *A*_{ij}, *B*_{ij} and *Γ*_{ij}, respectively, follow from the two different ways which can be used to derive the second-order equation of the system and the fact that the master symmetries commute (Xenitidis 2009). The latter implies that these equalities are identities. In fact, the basic identity is relation (4.8) from which equations (4.9, 4.10) follow. Specifically, the ∇_{i}-derivative of equation (4.8) leads to equation (4.9). On the other hand, multiplying the latter by *h*_{i} and taking the ∇_{i}-derivative of the resulting equation, one arrives at the third identity (4.10).

Some useful general properties of the operator ∇_{i} defined in equation (4.7) can be derived directly from its definition and are given in the following statement.

### Proposition 4.3

*Let h*_{i}(*u*,*u*_{i}), *h*_{j}(*u*,*u*_{j}) *be any pair of smooth functions and* ∇_{i} *be defined as in equation* (4.7). *Then the following identities hold*:

∇

_{i}*h*_{i}=0*identically*;[∇

_{i},∇_{j}]*X*=0,*for any function X*;*for any function Y independent of u*.

An important property of *Σ*_{ij} is its multidimensional consistency, i.e. the compatibility conditions among the equations of systems *Σ*_{ij}, *Σ*_{jℓ} and *Σ*_{ℓi} lead to no further restrictions. To make more precise what we mean by a multidimensionally consistent system of PDEs, we give the following definition in a way parallel to definition 2.1 of §2 for the discrete case.

### Definition 4.4

Let *F*(*u*,∂_{αi}*u*,∂_{αj}*u*,∂_{αi}∂_{αj}*u*)=0 denote a system of differential equations. Imposing the dependence of *u* on a third continuous variable *α*_{k} such that
if the three different ways to evaluate ∂_{αi}∂_{αj}∂_{αk}*u* lead to the same result, then we call the system of PDEs defined by *F* multidimensionally consistent.

The multidimensional consistency of *Σ*_{ij} can be checked by straightforward calculations and follows from the identity
4.11
which can be checked case by case for all the ABS equations and their corresponding functions *H*_{ij}.

To see how identity (4.11) is sufficient for the multidimensional consistency of *Σ*_{ij}, consider the compatibility condition
which must hold on the solutions of *Σ*_{ij}, *Σ*_{jℓ} and *Σ*_{ℓi}. First we use the equations of *Σ*_{ij} and *Σ*_{ℓi} to substitute the first-order derivatives of *u*_{i} in the above compatibility condition and then write it out explicitly. Next, we use the equations of *Σ*_{ij}, *Σ*_{jℓ} and *Σ*_{ℓi} to substitute the derivatives of *u*, *u*_{i}, *u*_{j} and *u*_{ℓ} involved in the resulting expression. After this, one arrives at
which holds in view of equation (4.11). In the same fashion, the three different ways to evaluate ∂_{αi}∂_{αj}∂_{αk}*u* and their consistency lead to similar expressions which are satisfied whenever *G*_{ijℓ}≡0. In the following section where Lagrangian functions will be constructed for the system *Σ*_{ij} we will see that the latter identity also ensures that these Lagrangians satisfy the closure relation.

## 5. Lagrangian formulation of the differential–difference and PDE system

In this section, we will show that there exist Lagrangian structures for the differential–difference equations (4.2) and for the PDE system *Σ*_{ij}, where in addition the Lagrangian of the latter satisfies a closure relation. We will start by presenting the Lagrangian for the system defined by the differential–difference equation (4.2).

### Proposition 5.1

*The following action functional*
5.1
*forms an action for the differential*–*difference system* (4.2) *in the sense that the EL equation*
5.2
*holds if u*=*u*(*n*_{i},*α*_{i}) *is a solution of system* (4.2).

### Proof.

The proof is by direct computation, and uses crucially the relation (3.10), as well as the identity
which holds for general symmetric bi-quadratics *h*_{i}. Thus, writing out explicitly EL equation (5.2), it takes a form which is a combination of two copies of the differential–difference equation (4.2) one shifted forward, and one shifted backwards in the lattice variable *n*_{i}. ■

Identifying the multidimensional consistency of *Σ*_{ij} with relation (4.11), the consistency property implies the closure relation for the corresponding Lagrangian 2-form structure as follows from the proof of the following theorem.

### Theorem 5.2

*A Lagrangian of system Σ*_{ij} *is given by
*
5.3
*and is antisymmetric:* *.*

*Moreover, Lagrangians* *and* *satisfy the closure relation
*
5.4
*on solutions of systems Σ*_{ij}*, Σ*_{jℓ} *and Σ*_{ℓi}*.*

### Remark 5.3

are functions which can be regarded as defining the components of the Lagrangian two-form . In this context, the geometric meaning of equation (5.4) is that is closed in the three-dimensional space subject to the equations of motion, i.e. on solutions of *Σ*'s.

### Proof.

The variational derivatives of the action with respect to *u*_{i} and *u*_{j} yield the second equation of *Σ*_{ij}, while its variational derivative with respect to *u* leads to a linear combination of the two first equations of *Σ*_{ij}. Indeed, consider the variational derivative of the action with respect to *u*_{i}, i.e.
Writing out explicitly the above relation, the terms involving the derivative of *u*_{i} with respect to *α*_{j} cancel out. Factorizing the remaining terms, one arrives at
which is identically zero in view of the third equation of *Σ*_{ij}.

Similarly, the variational derivative of *S* with respect to *u*_{j} takes the form

For the variational derivative with respect to *u*, we have
The terms in the parentheses are the two first equations of the system, thus the above equation becomes
Writing out explicitly the right-hand side of the last relation, we use the two first equations of the system to substitute the derivatives of *u*_{i} and *u*_{j}. After these substitutions, the terms involving the derivatives of *u* cancel out and we arrive at
Using the third identity of proposition 4.3, one easily concludes that *δ*_{u}*S*=0.

Denoting by the left-hand side of equation (5.4), i.e.
we will prove that on solutions of *Σ*_{ij}, *Σ*_{jℓ} and *Σ*_{ℓi}.

First, we work out explicitly to show that the second-order derivatives of *u*_{i}, *u*_{j} and *u*_{ℓ} cancel out. Then, we factorize the remaining terms with respect to the derivatives of *u*, which leads to the following expression
5.5
where

The coefficient of ∂_{αi}∂_{αj}*u* in equation (5.5) vanishes by taking into account the first-order equations of *Σ*_{jℓ} and *Σ*_{ℓi}. Due to the symmetry of , the coefficients of the remaining second-order derivatives of *u* also vanish.

On the other hand, the terms in equation (5.5) involving only first-order derivatives of *u* take the following form:
Obviously, *G*_{ijℓ}≡0 implies that the above expression vanishes on the solutions of systems *Σ*_{ij}, *Σ*_{jℓ} and *Σ*_{ℓi} and, subsequently, the closure relation (5.4) is satisfied. ■

## Acknowledgements

P.X. is supported by the *Newton International Fellowship* grant NF082473 entitled ‘Symmetries and integrability of lattice equations and related partial differential equations’, which is run by The British Academy, The Royal Academy of Engineering and The Royal Society. S.L. was supported by the UK Engineering and Physical Sciences Research Council (EPSRC).

## Appendix A. The list of the ABS equations and their characteristic polynomials

The following constitutes the list of scalar affine-linear multidimensionally consistent quadrilateral lattice equations up to Möbius transformations that was established by Adler *et al.* (2003). The equations of the form A1, A2 of their paper are omitted as they are related to other members of the list by point transformations. The form of Q4 given below was established by Hietarinta (2005), in which *sn* denotes the Jacobi elliptic function *sn*(*x*|*k*) with modulus *k*.
H1
H2
H3
Q1
Q2
Q3
Q4

The formulae for the canonical bi-quadratics *h* and *H*, and the corresponding discriminant curves, for the ABS equations are collected in the following lists:
Discriminant curve : where .

## Appendix B. Proof of Lemma 3.1

The proof follows from the symmetry analysis of the ABS equations which does not rely on the parametrization of the latter. Thus, we present it here without assuming any particular parametrization.

As it was proved in Tongas *et al.* (2007), all of the ABS equations admit three-point generalized symmetries which have the following form:
Moreover, it was proved that any other three-point generalized symmetry *K* is necessarily of the form , where the functions *a*(*n*_{i}) and *ϕ*(*n*_{i},*n*_{j},*u*) are determined by the solutions of a linear equation. Additionally, it was proved that all of the ABS equations admit extended generalized symmetries of the form .

On the other hand, it was shown in Mikhailov *et al.* (2010) that a higher order generalized symmetry for all of the ABS equations is given by

The commutator of the extended symmetry *V*_{i} with the first generalized symmetry yields
which is another symmetry of the equation under consideration. Since *K*^{(2)} is a symmetry of the equation, the same must hold for the remaining part of the above commutator. This means that the latter has to be of the general form *K* which implies
But, for all of the ABS equations and the parametrization in which *ξ*(*x*)=1 (see appendix A), it can be verified case by case that *μ*≡0. Consequently, from and using a point transformation , one is led to with . For instance, one could use the parametrization used in Adler *et al.* (2003) to verify the latter relation, cf. Xenitidis (2009) where the corresponding functions *ξ* and *h*_{i} were given.

## Footnotes

↵1 It should be pointed out that only very special and non-trivial discretizations of a given integrable differential equation will retain the key integrability properties of the latter.

↵2 In fact, this constraint is equivalent to stating that the solutions obey a KP-type of equation for the embedding in a higher dimension, which holds true for known infinite families of solutions such as soliton solutions.

↵3 If all the polynomials are constants or separate variables, e.g.

*h*=*h*_{1}(*u*)*h*_{2}(*u*_{i}), obviously the arguments of these functions are functionally independent. In the generic case, let us denote by*X*and*Y*the first argument of*Φ*and*Ψ*, respectively, and assume that they are functionally dependent. This implies that*X*_{u}*Y*_{ui}=*X*_{ui}*Y*_{u}, from which after some differentiations follows that*h*_{ui}/*h*=*R*(*u*_{i},*u*_{j}). The latter implies that*h*separates variables, a contradiction.

- Received February 21, 2011.
- Accepted June 13, 2011.

- This journal is © 2011 The Royal Society