A Variational Principle for Discrete Integrable Systems

For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference equations with two independent variables, MDC allows us to define an action on arbitrary 2-dimensional surfaces embedded in a higher dimensional space of independent variables, where the action is not only a functional of the field variables but also the choice of surface. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also with respect to variations of the surface in the space of independent variables. Here we derive the resulting system of generalized Euler-Lagrange equations arising from that principle. We treat the case where the equations are 2 dimensional (but which due to MDC can be consistently embedded in higher-dimensional space), and show that they can be integrated to yield relations of quadrilateral type. We also derive the extended set of Euler-Lagrange equations for 3-dimensional systems, i.e., those for equations with 3 independent variables. The emerging point of view from this study is that the variational principle can be considered as the set of equations not only encoding the equations of motion but as the defining equations for the Lagrangians themselves.


Introduction
It is a notion going back to the 1600s that a dynamical system should minimize some quantity, i.e., that the equations of motion should arise as critical points of some action functional. This action is a quantity depending in principle on both the dependent and independent variables, and finding a minimum, or more generally a critical point, specifies a path followed by the dependent variable. The condition that specifies this path is the Euler-Lagrange equation.
A discrete calculus of variations was first developed outside the scope of integrable systems in the 1970s by Cadzow [6], Logan [11] and later Maeda [12,13,14]. Cadzow's original motivation was the use of the digital computer in modern systems and the solution of control problems, and it became clear that the formulation of a discrete calculus of variations was important for numerical methods, in optimization and engineering problems. In the discrete realm instead of the action being an integral of a Lagrangian, it is a sum over the independent variable(s).
In the case of multidimensionally consistent systems, we are able to embed the system in a higher-dimensional space, with compatible systems living in each subspace. Indeed, we may have an infinite number of compatible systems in an infinite number of dimensions, and we do not need to restrict to any particular subspace; we could have a system following a path through an arbitrary number of dimensions. So we now have to consider not only the path taken by the dependent variable(s) with respect to the independent variable(s), but also the path through this space of independent variables. Then it is natural to ask that the action be critical with respect to a change in the path of independent variables. This postulate was first put forward by the authors in [8], initially for 2-dimensional systems, both discrete and continuous. Requiring the action functional to be invariant under small changes in the path (which for 2-dimensional systems is a surface) through the space of independent variables leads to a condition on the Lagrangian, a closure relation, which was shown to be satisfied for many examples of multidimensionally consistent systems [8,9,2,10,17,5]. This serves as an answer to the question of how to encode an entire multidimensionally consistent system in a variational principle.
An issue with the usual variational principle is that it is often impossible to obtain the desired system of equations as Euler-Lagrange equations, but only an integrated or derived form of those equations. This can be seen in the continuous realm in the case of the potential Korteweg-de Vries (pKdV) equation, where the Euler-Lagrange equation gives a derivative of the pKdV; and it can be seen in the discrete realm in the case of quad equations, such as those in the Adler-Bobenko-Suris (ABS) classification [1], where the Euler-Lagrange equations give only consequences of the quad equations, which can be considered as discrete derivatives. One does not obtain a quad equation directly as an Euler-Lagrange equation on a fixed surface.
We show in this paper that the variational principle of [8], which considers variations of the surface, provides a set of Euler-Lagrange equations, specifying conditions on both the Lagrangian and Euler-Lagrange equations. In the 2-dimensional discrete case this set is enough to specify an equation on a single quad. The key point we wish to make is that the variational principle should be considered as supplying Lagrangians as solutions of a system of equations, as much as the equations of motion themselves. It is the latter perspective, invited by the phenomenon of multidimensional consistency as the defining aspect of integrability, that forms the main departure of our new variational principle from any of the conventional variational theories.
The case of 1-dimensional systems was examined in [19,18] and subsequently from the Hamiltonian perspective in [16,3]. Further work has also appeared recently on 2-dimensional systems in [4].
This paper is concerned with discrete systems. In Section 2 we examine the variational principle for 2-dimensional discrete systems: defining the action, listing the Euler-Lagrange equations for the basic configurations in the surface, and deriving quad equations as consequences of these Euler-Lagrange equations. We give examples of H1 and H3 to serve as illustrations. In Section 3 we give the defining set of Euler-Lagrange equations for 3-dimensional discrete systems, and show that these are compatible with the bilinear discrete Kadomsev-Petviashvili (KP) equation. Section 4 provides some further discussion and perspectives.  To this end, consider the surface σ to be a connected configuration of elementary plaquettes σij (n), where σij(n) is specified by the position n = (n, n + ei, n + ej) of one of its vertices in the lattice and the lattice directions given by the base vectors ei, ej, as in Figure 1. The surface can be closed, or have a fixed boundary. Since the 3-point Lagrangians depend on two directions in the lattice, and when embedded in a multidimensional lattice at each point can be associated with an oriented plaquette σij (n), we can think of these Lagrangians as defining a discrete 2-form Lij (n) whose evaluation on that plaquette is given by the Lagrangian function as follows Lij (n) = L(u(n), u(n + ei), u(n + ej ); αi, αj). (2.1) The Lagrangians given in [8] are all antisymmetric with respect to the interchange of lattice directions i, j, and so this is well-defined. Then the action S is also well-defined by Note that in performing this sum we must be careful to take into account the orientation of the plaquettes.

The Euler-Lagrange equations
To derive the set of Euler-Lagrange equations stemming from the action (2.2), we look at what happens at a particular point n in the lattice. For ease of notation we will suppress the dependence on n, writing u = u(n), and make use of shift operators Ti, writing Tiu = u(n + ei), Tj u = u(n + ej), T −1 i u = u(n − ei), etc. The postulate is that the system lies at a critical point of the action, and our point of view is that it lies at a critical point with respect to not only the dependent variable u, but also the independent variables, i.e., the surface σ. Since we are considering discrete surfaces here, the notion of infinitesimal variations of the independent variables does not make sense, and we can make only finite variations. Thus our postulate is that the action is independent of σ (keeping any boundary fixed) on solutions to the system.
It suffices to consider a collection of fixed surfaces embedded in 3 dimensions, and compute variations with respect to u on that surface. For an action which is the sum of 3-point Lagrangians  This corresponds to the Euler-Lagrange equation The elementary configurations in 3 dimensions are shown in Figure 3; all other configurations can be obtained as combinations of these. A statement to this effect appears in [4].
Note that in the final picture in Figure 3, only two plaquettes contribute, because of the 3-point nature of the Lagrangians we are considering here.
Each of these pictures corresponds to a different Euler-Lagrange equation. Since all surfaces in the lattice can be obtained by combining these elementary configurations, the Euler-Lagrange equation for any surface can be obtained by combining the Euler-Lagrange equations corresponding to the respective elementary configurations.

Theorem 1 The following form a complete set of Euler-Lagrange equations for the quadrilateral
for all i, j, k ∈ I, where I is the index set labelling the lattice directions.

Consequences of the Euler-Lagrange equations
As in the previous subsection, we consider actions which are the sum of 3-point Lagrangians L(u, Tiu, Tju; αi, αj ), where the Lagrangians are anti-symmetric with respect to the interchange of the lattice directions, so that the equations (2.4a)-(2.4c) hold. The one further assumption we will make is that we may choose initial conditions u, Tiu, Tju, T k u independently and arbitrarily. If we impose that the action remains invariant under perturbations of the surface, then it is independent of the surface [8], and all of these equations must hold simultaneously. Note that (2.3) is a consequence of (2.4a)-(2.4c) and their cyclic permutations. for some function C, which is (2.7). Note that since L(u, Tiu, Tju; αi, αj ) is antisymmetric under the interchange of lattice directions i, j, then the same must be true of C(Tiu, Tju; αi, αj ). where h(u) is an arbitrary function, which can be absorbed into A.
Proof: Substituting (2.7) into equations (2.4b) and (2.4c) gives where we have already cancelled some of the terms (provided we assume that Tj u and T k u can be independently chosen, so that they don't depend on u). We see that we can rewrite these in a suggestive way, isolating dependence on particular lattice directions: and of course this must be true for all i, j, k. Thus we must have where ∆i is a difference operator defined by ∆i = Ti − id.

Example: H1
If we consider the example of H1, the Lagrangian (which was first given in [7]) evaluated on a plaquette in the (i, j)-direction has the form which consists of 2 shifted copies of H1 lying on a 7-point configuration, i.e., a consequence of H1. The Euler-Lagrange equations on non-flat surfaces (2.4a)-(2.4c) are respectively In fact, (2.21d) is a consequence of (2.21c) and its copies under permutation of lattice directions. Also equation (2.12) with h taken to be zero is which is consistent around a cube for arbitrary λ.
(2.37) Therefore ti = tj, and if δ = 0 then ti = tj = 1, and we have the usual equation H3. If on the other hand δ = 0 we have a little more freedom, and we can let ti = tj = t for some arbitrary constant t. In that case, the equation is αi(uTiu + tTjuTiTju) − αj (uTju + tTiuTiTju) = 0, (2.38) and this equation is also consistent around the cube.

Defining the action
A Lagrangian for a 3-dimensional system can be defined on an elementary cube ν ijk (n), where ν ijk (n) is specified by the position n = (n, n + ei, n + ej, n + e k ) of one of its vertices in the lattice and the lattice directions given by the base vectors ei, ej, e k , as in Figure 4. The Lagrangian can depend in principle on the fields at all 8 vertices of the elementary cube: L ijk (n) = L(u(n), u(n+ei), u(n+ej ), u(n+e k ), u(n+ei+ej ), u(n+ej+e k ), u(n+ei+e k ), u(n+ei+ej+e k )).

The Euler-Lagrange equations
The Euler-Lagrange equation in the usual 3-dimensional space is where we take into account all Lagrangian contributions that involve the field u. We have suppressed the dependence on the variables, writing L ijk = L ijk (u, Tiu, Tju, T k u, TiTju, TjT k u, TiT k u, TiTjT k u).
Note that any point in Z 3 belongs to 8 cubes, so we have in principle 8 terms in the above equation. This is the analogue of the "flat" equation (2.3) we had in 2 dimensions. Embed the system in 4 dimensions. In 3 dimensions, the smallest closed 2-dimensional space is a cube, consisting of 6 faces; in 4 dimensions, the smallest closed 3-dimensional space is a hypercube, consisting of 8 cubes. The action on the elementary hypercube will have the form Because of the symmetry, we need only take derivatives with respect to u, Tiu, TiTju, TiTj T k u and TiTjT k T l u, and the other equations will follow by cyclic permutation of the lattice directions. Then we have the set of equations along with the equivalent shifted versions (3.6e)

Example: bilinear discrete KP
The Lagrangian for the bilinear discrete KP equation was first given in [9], and in 3-dimensional space gives as Euler-Lagrange equations 12 copies of the bilinear discrete KP equation itself, on 6 elementary cubes. The Lagrangian L ijk depends on the six fields Tiu, Tj u, T k u, TiTj u, TjT k u, and TiT k u, and has the following explicit form: − ln TiTju ln TjT k u − ln Tj T k u ln T k Tiu − ln T k Tiu ln TiTju + ln Tiu ln Tj u + ln Tj u ln T k u + ln T k u ln Tiu Here the Aij are constants which are antisymmetric with respect to swapping the indices. If we introduce the quantity C ijk , defined by then the bilinear discrete KP equation itself can be written C ijk = 1. The usual Euler-Lagrange equation is The Euler-Lagrange equations (3.5a) and (3.5e) are trivial in this case, while (3.5b)-(3.5d) are

Summary and conclusions
Multidimensionally consistent systems can be considered as critical points of an action: critical with respect to the dependent variable, and also with respect to the curve or surface in the space of independent variables. In the case of discrete systems, this means the action is required to be independent of the curve or surface on which it is defined, whilst keeping any boundary it may have fixed. This leads to a set of Euler-Lagrange equations, corresponding to basic configurations of points in a surface, which should be satisfied simultaneously.
In the case of 2-dimensional discrete systems, we have shown that the set of Euler-Lagrange equations arising from this variational principle specify firstly a particular form of the Lagrangian, and furthermore quad equations themselves, whereas previously only a weaker form of the equations could be derived. Starting from known examples of Lagrangians, we can show that the resulting quad equations are compatible with previous results.
It would be interesting to see if the results of this paper can be extended to higher than 3 dimensions where we will have a Lagrangian function evaluated on an n-dimensional object, in particular on an n-dimensional cube. Embedding this in higher dimensions, we consider an action on the smallest closed n-dimensional surface in (n+1) dimensions, a hypercube. Then the minimal set of Euler-Lagrange equations are obtained by demanding that the derivative of this action with respect to each variable is zero.
As we pointed out earlier, the set of Euler-Lagrange equations could, and maybe should, be viewed as a system of equations for the Lagrangian itself. This constitutes a significant departure from the conventional point of view where the Lagrangian is a given object (usually obtained from considerations of physics) and the main issue is to derive the equations of the motion of the system from a variational approach. In the integrable case of Lagrangian multiforms, the Lagrangians themselves are part of the solution of the extended system of equations obtained from varying not only the field variables on a given space-time of independent variables, but by also varying the geometry of space-time itself. It would be of interest to see whether Lagrangians associated with descriptions of known physical processes could be obtained from such a novel variational theory.