A variational perspective on continuum limits of ABS and lattice GD equations

A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand-Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures where previously unknown. This includes the Krichever-Novikov hierarchy, the double hierarchy of Sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system.


Introduction
This paper continues the work started in [23], where we established a continuum limit procedure for lattice equations with a pluri-Lagrangian (also called "Lagrangian multiform") structure. Here, we apply this procedure to many more examples. These continuum limits produce known hierarchies of integrable PDEs, but also a pluri-Lagrangian structure for these hierarchies, which in most cases was not previously known.
We start by giving a short introduction to discrete and continuous pluri-Lagrangian systems. Then, in Section 2, we review the continuum limit procedure from [23]. In Sections 3 and 4 we present examples form the ABS list [3]. In Section 5 we extend one of these examples, ABS equation H3, to produce the doubly infinite hierarchy containing the Sine-Gordon and modified KdV equations. In section 6 we comment on a common feature of all of our continuum limits of ABS equations, namely that half of the continuous independent variables can be disregarded. In Section 7 we study two examples from the Gelfand-Dickey hierarchy.
The computations in this paper were performed in the SageMath software system [21]. The code is available at https://github.com/mvermeeren/pluri-lagrangian-clim.  ABS list. In the same paper Lagrangians were given for each of the equations in the context of the classical variational principle. We now present the pluri-Lagrangian perspective on these equations, which first appeared in [13] and was explored further in [4]. Consider the lattice Z N with basis vectors e 1 , . . . , e N . To each lattice direction we associate a parameter λ i ∈ C. We denote an elementary square (a quad ) in the lattice by i,j (n) = n + ε 1 e i + ε 2 e j ε 1 , ε 2 ∈ {0, 1} ⊂ Z N , where n = (n 1 , . . . , n N ). Quads are considered to be oriented; interchanging the indices i and j reverses its orientation. We will write U ( i,j (n)) for the quadruple U ( i,j (n)) = U (n), U (n + e i ), U (n + e j ), U (n + e i + e j ) .
Occasionally we will also consider the corresponding "filled-in" squares in R N , i,j (n) = n + µ 1 e i + µ 2 e j µ 1 , µ 2 ∈ [0, 1] ⊂ R N , on which we consider the orientation defined by the 2-form dt i ∧ dt j .
The role of a Lagrange function is played by a discrete 2-form L(U ( i,j (n)), λ i , λ j ), which is a function of the values of the field U : Z N → C on a quad and of the corresponding lattice parameters, where L(U ( ji (n)) , λ j , λ i ) = −L( ij (n), λ i , λ j ) . Consider a discrete surface Γ = { α } in the lattice, i.e. a set of quads, such that the union of the corresponding filled-in squares α α is an oriented topological 2-manifold (possibly with boundary). The action over Γ is given by Definition 2. The field U is a solution to the discrete pluri-Lagrangian problem if it is a critical point of S Γ (with respect to variations that are zero on the boundary of Γ) for all discrete surfaces Γ simultaneously.
Euler-Lagrange equations in the discrete pluri-Lagrangian setting are obtained by taking point-wise variations of the field on the corners of an elementary cube. Since any other surface can be constructed out of such corners, these variations give us necessary and sufficient conditions.
It is useful to note that exact forms are null Lagrangians.
Proposition 3. Let η(U, U i , λ i ) be a discrete 1-form. Then every field U : Z N → C is critical for the discrete pluri-Lagrangian 2-form L = ∆η, defined by Proof. The action sum of L = ∆η over a discrete surface Γ depends only on the values of U on the boundary of Γ. Hence any variation that is zero on the boundary leaves the action invariant.
For the details of the discrete pluri-Lagrangian theory we refer to the groundbreaking paper [13], which introduce the pluri-Lagrangian (or Lagrangian multiform) idea, and to the reviews [5] and [10,Chapter 12], and the references therein.

The continuous pluri-Lagrangian principle
Continuous pluri-Lagrangian systems are defined analogous to their discrete counterparts. Let L[u] be a 2-form in R N , depending on a field u : R N → C and any number of its derivatives. The first question about continuous pluri-Lagrangian systems is to find a set of equations characterizing criticality in the pluri-Lagrangian sense. If L = i<j L ij [u] dt i ∧dt j , then these equations are for i, j, and k distinct, where and u I denotes a partial derivative corresponding to the multi-index I. We write that I t k if the entry in I corresponding to t k is nonzero, i.e. if at least one derivative is taken with respect to t k , and I t k otherwise. Compared to u I , the field u It α i t β j has α additional derivatives with respect to t i and β additional derivatives with respect to t j .

Continuum limit procedure
In [23] we presented a continuum limit procedure for multidimensionally consistent lattice equations. Here we review the essentials of this procedure.

Miwa variables
We construct a map from the lattice Z N (n 1 , . . . , n N ) into the continuous multi-time R N (t 1 , . . . , t N ) as follows. We associate a parameter λ i with each lattice direction and set Note that a single step in the lattice (changing one n j ) affects all the times t i , hence we are dealing with a very skew embedding of the lattice. We will also consider a slightly more general correspondence, for constants c, τ 1 , . . . , τ N describing a scaling and a shift of the lattice. The variables n j and λ j are known in the literature as Miwa variables and have their origin in [16]. We will call Equation (4) the Miwa correspondence.
We denote the shift of U in the i-th lattice direction by U i . If U (n) = u(t 1 , . . . , t N ), It is given by which we can expand as a power series in λ i . The difference equation thus turns into a power series in the lattice parameters. If all goes well, its coefficients will define differential equations that form an integrable hierarchy. Generically however, we will find only t 1 -derivatives in the leading order, because only in the t 1 -coordinate do the parameters λ i enter linearly. To get a hierarchy of PDEs, we need to have some leading order cancellation, such that the first nontrivial equation contains derivatives with respect to several times. Given an integrable difference equation, it is a nontrivial task to find an equivalent equation that yields the required leading order cancellation. Note that such a procedure is strictly speaking not a continuum limit; sending λ i → 0 would only leave the leading order term of the power series. A more precise formulation is that the continuous u interpolates the discrete U for sufficiently small values of λ i , where U is defined on a mesh that is embedded in R N using the Miwa correspondence. Since λ i is still assumed to be small, it makes sense to think of the outcome as a limit, but it is important to keep in mind that higher order terms should not be disregarded.

Continuum limits of Lagrangian forms
We sketch the limit procedure for pluri-Lagrangian structures introduced in [23]. Consider N pairwise distinct lattice parameters λ 1 , . . . , λ N and denote by e 1 , . . . , e N the unit vectors in the lattice Z N . The differential of the Miwa correspondence maps them to linearly independent vectors in R N : We start from a discrete Lagrangian two-form L disc . A discrete field can be recovered from a continuous field by evaluating it at u(t), where the differential operators correspond to the lattice directions under the Miwa correspondence, As the subscript indicates, L disc is not yet a continuous Lagrangian, because the action is a sum over evaluations of L disc on a corner of each quad of the surface. Using the Euler-Maclaurin formula we can turn the action sum into an integral of which is a formal power series in λ 1 and λ 2 . Then by construction we have the formal equality where (η 1 , . . . , η N ) are the 1-forms dual tot the Miwa shifts (v 1 , . . . , v N ) and 12 is the embedding under the Miwa correspondence of the filled-in square. Since we want such a relation on arbitrarily oriented quads, we consider the continuous 2-form This is the continuum limit of the discrete pluri-Lagrangian 2-form, but it can be written in a more convenient form in terms of the coefficients of the power series L Miwa .
Theorem 5. Let L disc be a discrete Lagrangian 2-form, such that every term in the corresponding power series L Miwa is of strictly positive degree in each λ i , i.e. such that L Miwa is of the form Then the differential 2-form is a pluri-Lagrangian structure for the continuum limit hierarchy, restricted to R N .

Eliminating alien derivatives
Suppose a pluri-Lagrangian 2-form in R N produces multi-time Euler-Lagrange equations of evolutionary type, If this is the case, then the differential consequences of the multi-time Euler-Lagrange equations can be written in a similar form, where the multi-index I in f I is a label, not a partial derivative. In this context it is natural to consider t 1 as a space coordinate and the others as time coordinates. If the multi-time Euler-Lagrange equations are not evolutionary, Equation (6) still holds for a reduced set of multi-indices I.

Definition 6.
A mixed partial derivative u I is called {i, j}-native if each individual derivative is taken with respect to one of the t i , . . . , t j or with respect to the space coordinates t 1 , i.e. if I t k ⇒ k ∈ {1, i, j}.
If u I is not {i, , j}-native, i.e. if there is a k ∈ {1, i, j} such that t k ∈ I, then we say u I is {i, j}-alien.
If it is clear what the relevant indices are, for example when discussing a coefficient L i,j , we will use native and alien without mentioning the indices.
We would like the coefficient L i,j to contain only native derivatives. A naive approach would be to use the multi-time Euler-Lagrange equations (6) to eliminate alien derivatives. Let R i,j denote the operator that replaces all {i, j}-alien derivatives for which the multi-time Euler-Lagrange equations provide an expression. We denote the resulting pluri-Lagrangian coefficients by L i,j = R i,j (L i,j ) and the 2-form with these coefficients by L. A priori there is no reason to believe that the 2-form L will be equivalent to the original pluri-Lagrangian 2-form L, but using the multi-time Euler-Lagrange equations one can derive the following result [23].
Theorem 7. If for all j the coefficient L 1j [u] does not contain any alien derivatives, then every critical field u for the pluri-Lagrangian 2-form L is also critical for L.
In practice, given a Lagrangian 2-form, we can make it fulfill the conditions on the L 1j [u] by adding a suitable exact form and adding terms that attain a double zero on solutions to the multi-time Euler-Lagrange equations. Neither of these actions affects the multi-time Euler-Lagrange equations.

ABS equations of type Q
All equations of type Q can be prepared for the continuum limit in the same way, based on their particularly symmetric three leg form. In Section 3.1 we present this general strategy. In Sections 3.2-3.7 we will discuss each individual equation of type Q.

Three leg forms and Lagrangians
All quad equations Q(V, V 1 , V 2 , V 12 , α 1 , α 2 ) = 0 from the ABS list have a three leg form: For the equations of type Q, the function Φ on the long (diagonal) leg is the same as the function Ψ on the short legs: where we introduced new parameters by λ 2 i = α i . Suitable leg functions Ψ were listed in [4]. For the purposes of a continuum limit, it is useful to reverse one of the time directions, i.e. to consider and to write the Ψ in terms of difference quotients. We will introduce a function of the continuous variables, from which we can recover Ψ(V, W, λµ) by plugging in suitable approximations to v and v . Note that Ψ takes only one parameter, which is the product of the two parameters of ψ. For all of the ABS equations we will use c = −2 in the Miwa correspondence, which means that the derivative v t 1 is approximated by difference quotients such as All equations of the Q-list can be written in the form To achieve the symmetries required of a quad equation, we require that Furthermore, we require that ψ(v, v , 0, 0) = 0. We would expect the first nonzero terms in the series expansion at first order in λ 1 , λ 2 , but one can easily check that This is the leading order cancellation required to obtain PDEs in the continuum limit: at the first order, where generically we would get only derivatives with respect to t 1 , we get nothing at all. Equation (7) also reveals a reason for considering the three-leg form with a "downward" diagonal leg, as in Figure 3 can be expanded in a double power series, but its "upward" analogue To find a Lagrangian for Equation (7), we follow [3,4] and integrate the leg function ψ. We take Then gives the terms of the Lagrangian in triangle form: Note the symmetries of Λ: In some cases we will rescale Λ and hence L by a constant factor. This is purely for esthetic reasons and does not affect the multi-time Euler-Lagrange equations.
Proposition 8. Solutions to the quad equation Q = 0 in the plane, with Q given by Equation (7), are critical fields for the action of the Lagrangian given by Equation (8).
Proof. We have Similarly, we have Summing up all derivatives of the action in the plane with respect to the field at one vertex, and using the symmetry of the quad equation, we find two shifted copies of Equation (7), arranged as in Figure 3(d).
The Lagrangian constructed this way is suitable for the continuum limit procedure, as the following proposition establishes.
Proposition 9. Every term of L is of at least first order in both parameters, L = O(λ 1 λ 2 ).

Q1 δ=0
A continuum limit for equation Q1, with its pluri-Lagrangian structure, was given in [23]. The results is the hierarchy of Schwarzian KdV equations, where v i is a shorthand notation for the derivative v t i . Table 1 shows how this continuum limit fits in the scheme of Section 3.1. We find the discrete Lagrangian Table 1: Q1 δ=0 fact sheet. See Section 3.1 for the meaning of these functions.
The first few coefficients of the continuous pluri-Lagrangian 2-form, after eliminating alien derivatives, are The even-numbered times correspond to trivial equations, v t 2i = 0, restricting the dynamics to a space of half the dimension. We can also restrict the pluri-Lagrangian formulation to this space: is a pluri-Lagrangian 2-form for the hierarchy of nontrivial Schwarzian KdV equations,

Q1 δ=1
Equation Q1 δ=1 reads We apply the procedure of Section 3.1 to find a Lagrangian that is suitable for the continuum limit. Intermediate steps are listed in Table 2. Table 2: Q1 δ=1 fact sheet. See Section 3.1 for the meaning of these functions.
In the continuum limit of the equation we find and v 2k = 0 for all k ∈ N. Some coefficients of the continuous pluri-Lagrangian 2-form are

Q2
For the equation the general strategy of Section 3.1 works with the choices listed in Table 3. Table 3: Q2 fact sheet. See Section 3.1 for the meaning of these functions.
The continuum limit hierarchy is . . . and v 2k = 0 for all k ∈ N. A few coefficients of the pluri-Lagrangian 2-form are We consider equation Q3 in the form The strategy of Section 3.1 works as outlined in Table 4.
. . . and v 2k = 0 for all k ∈ N. Some coefficients of the continuous pluri-Lagrangian 2-form are Note on dilogarithms. In Table 4 we encounter the dilogarithm function [25], given by for |z|< 1 and by analytic continuation for other z ∈ C \ [1, ∞). Its derivative is This can be used to integrate log • sin, which occurs in the leg function Ψ, log(sin z)) dz = i 2 −z 2 + Li 2 e 2iz − z log(−2i) + c.
In the continuum limit procure we need a series expansion of Li 2 (e x ). For small x this seems problematic, since Li 2 (z) has a branch point at z = 1. The way to handle this is to carry along this branch point in the formal series expansion where the first term famously equals Li 2 (1) = π 2 6 .

Q3 δ=1
For equation Q3 with nonzero parameter we use a slightly different form compared to Q3 δ=1 , The general strategy of Section 3.1 applies to this transformed equation with the choices in Table 5.
. . . and v 2k = 0 for all k ∈ N. A few coefficients of the continuous pluri-Lagrangian 2-form are

Q4
Consider equation Q4 in its elliptic form, , and ℘ is the Weierstrass elliptic function. The approach of Section 3.1 works as outlined in Table 6. A few additional functions and constants appear in this table: the Weierstrass functions σ and ζ, with the same periods as ℘, and the invariants g 2 and g 3 of the ℘function. They satisfy Additionally, for the series expansion of ψ we use where the first coefficients are a 0,0 = 1, a 1,0 = −1, and a 0,1 = −3. These and many more identities involving the Weierstrass elliptic functions can be found in [1, Chapter 18].
The continuum limit hierarchy is and v 2k = 0 for all k ∈ N. To simplify these equations we used the doubling formula for the Weierstrass function, Table 6: Q4 fact sheet. See Section 3.1 for the meaning of these functions.
As the first nontrivial equation of the limit hierarchy we recognize the Krichever-Novikov equation [12], [18]. The continuum limit equations found in all previous example can be considered as degenerations of the KN equation and its hierarchy, just like all other ABS equations of type Q can be obtained from Q4 [2].
The continuous pluri-Lagrangian 2-form has the coefficients

ABS equations of type H
For quad equations of type H we do not have a general strategy to find a form of the difference equation and its Lagrangian that is suitable for the continuum limit. This has to be investigated on a case-by-case basis. The continuum limit of equation H1 was discussed in [23].

H2
No continuum limit is known for equation H2, which reads We give a heuristic explanation of the obstruction one encounters when trying to pass to the continuum limit. A tempting trick would be to change the sign of the field at every other vertex, V (n, m) = (−1) n+m U (n, m). The resulting equation has a suitable power series expansion, but the sign of the second term depends on the location in the lattice. In other words, the equation has become nonautonomous. Though there should not be any fundamental objection to this, the pluri-Lagrangian theory for nonautonomous systems has not yet been developed.
Of course the fact that one particular change of variables fails does not imply that there is no continuum limit. A better perspective on the issue is given by background solutions. When the zero field U ≡ 0 is a solution to the quad equation, the continuum limit assumes the field to be small compared to the inverse of the parameters. If U ≡ 0 is not a solution, as is the case for Equation (11), one can look for a different solution to expand around. Such solutions are known as background solutions. There are two kinds of background solutions for H2 that treat both lattice directions equally [9]: Expanding around either of these solutions is equivalent to expanding around 0 after a nonautonomous changes of variables. As in our first attempt above, these changes of variables make the equation nonautonomous.

H3 δ=0
Equation H3 is usually given as but it is useful to perform a nonautonomous transformation U (n, m) → i n+m U (n, m), which gives us which is known as the lattice modified KdV equation [17]. The continuum limit of Equation (13) is the hierarchy . . . and u 2k = 0 for all k ∈ N. However, we run in to trouble on the Lagrangian level. The Lagrangian given in [13] for H3, adapted to the case δ = 0, is where Li 2 is the dilogarithm function. Unfortunately, the occurrence of expressions like log(λ i ) prohibits a power series expansion in the parameters λ i . In order to find a better form of H3 δ=0 , with an expandable Lagrangian, we make the transformation U = exp i 2 V . Then Equation (13) turns in to Multiplying with exp − i 4 (V + V 1 + V 2 + V 12 ) turns this into which is the form in which H3 δ=0 arises from Bcklund transformations for the Sine Gordon equation [10, p60]. Additionally, it is only a simple transformation removed from the discrete Sine-Gordon equation of Hirota [11]. Using standard trigonometric identities we can rewrite the equation as which we can put in a three-leg form: From the three-leg form we can derive a Lagrangian as in [4]. It takes the implicit form There is no need to evaluate this integral exactly. Instead one can expand the integrand as a power series up to any desired order in x and integrate this series. This is sufficient to write L as a power series up to the corresponding order in the parameters λ i . A leading order calculation shows that this power series does not contain any terms of nonpositive order in either of the parameters, so it is suitable for the continuum limit.
In the continuum limit of Equation (14) we find the potential modified KdV hierarchy, and v 2k = 0 for all k ∈ N. Some coefficients of the pluri-Lagrangian 2-form are Due to difficulties analogous to those of equation H2, no continuum limit is known for equation H3 δ=1 ,

H3 and the Sine-Gordon equation
Equation H3 is not just related to the modified KdV equation, but also to the Sine-Gordon equation. In this section we investigate how to find both in the continuum limit.
To do so we need to transform Equation (13) in a way that breaks the symmetry: Setting again U = exp i 2 V , we find the equation In order to recover some kind of multidimensional consistency for this equation, we consider an even-dimensional lattice Z 2N with coordinates (n 1 , . . . , n N , n 1 , . . . , n N ), where the coordinates (n 1 , . . . , n N ) correspond to lattice directions where the above transformation has taken place. Then on any quad spanned by directions n k and n we impose Equation (15), The original equation H3, in the form of Equation (14), is imposed on quads spanned by directions n k and n , and also on quads spanned by directions n k and n , because transforming both directions leaves the equation invariant.
We extend the Miwa correspondence, treating the barred variables in the same way as the variables without decoration, but contributing to a separate set of continuous variables (t 1 , . . . , t N ), Then in the leading order of the continuum limit, Equation (16) produces the Sine-Gordon equation, In addition to this equation, the full series expansion in the parameters λ k and λ yields two copies of the modified KdV hierarchy, These are the same hierarchies we obtain when taking the continuum limits of the equations imposed on quads spanned by directions n k and n , respectively n k and n .
The discrete Lagrangian takes slightly different forms on the three types of quads: To get the required leading-order cancellation, we add to L the exterior derivative of the discrete 1-form η given by η(V, V ) = − 1 8 (V 2 + V 2 ) and η(V, V k ) = 0. This yields The resulting pluri-Lagrangian 2-form is of the form where L ij = −L ji . If i, j ∈ {1, 2, . . .}, then the coefficients are exactly those found in Section 4.2. If i, j ∈ {1, 2, . . .} the coefficients are obtained from those by placing a bar over every derivative and putting a minus sign in front. As before, we can restrict our attention to the odd-numbered times to get a pluri-Lagrangian structure for the nontrivial equations, Some coefficients of mixed type are A few examples of multi-time Euler-Lagrange equations are One recognizes the equations of the double hierarchy. All multi-time Euler-Lagrange equations are either trivial, or differential consequences of the equations of the hierarchy.

About the even-numbered times
In all the examples so far, only the odd-numbered times have nontrivial equations associated to them. This is consistent with several purely continuous descriptions of integrable hierarchies. From the perspective of continuum limits, it follows from the fact that for all equations we dealt with, Q is an even or odd function of the parameters.
Proof. Due to the symmetry assumptions on Q, we have Let u(t 1 , t 2 , . . .) be a solution to the continuum limit equations. Then Comparing Equations (17) and (18), we immediately see that their series expansions in λ 1 , λ 2 respectively µ 1 , µ 2 only differ by a minus sign for each derivative with respect to an even-numbered time t 2k . In other words, if we have a solution u to the continuum limit hierarchy, then reversing all even times gives a new solution.
Corollary 11. If the continuum limit hierarchy of the difference equation Q = 0, as in Proposition 10, consists of evolutionary equations u k = f k (u, u 1 , u 11 , . . .), then for all even k we have f k = 0.
Unfortunately, as we will see in in Section 7, there are situations where the continuum limit contains PDEs that are not evolutionary. Here we will see a different pattern of trivial equations. This pattern can be understood in the framework of reductions of the lattice KP system [7,8], or from the construction by pseudodifferential operators of the Gelfand-Dickey hierarchy [6].

Gelfand-Dickey hierarchies
A discrete counterpart of the Gelfand-Dickey hierarchy was introduced in [19], its pluri-Lagrangian structure in [14]. The N -th member of the discrete hierarchy is a system of quad equations with 2N − 3 components, which we denote by V [1] , . . . , for j = 0, . . . , N − 3, and where It is important to note that the variables V [N −2] and W [N −2] can be eliminated. The reduced system consists of equations (19)- (20) for j = 0, . . . , N − 4 and the 9-point equation obtained by evaluating V once with Equations (19)-(20) and once with Equation (21).
The Lagrangian given in [14] for the N -th lattice Gelfand-Dickey equation is Note that this Lagrangian depends on the field V [N −2] but not on W [N −2] . A more symmetric equivalent Lagrangian is obtained by adding the exact discrete differential   (21), but rather the 9-point version (22). In particular this means that, from the Lagrangian perspective, the scalar form of the Boussinesq equation (N = 3) is the most natural. The first truly multi-component Lagrangian equation of the hierarchy is found for N = 4. The continuum limit of this equation will allow us to formulate a continuous multi-component pluri-Lagrangian system. Note that the Lagrangian (23) depends on fields on a single quad and on the corresponding lattice parameters. Hence it fits the pluri-Lagrangian theory. The multidimensional consistency of the equations can either be checked as consistency around an elementary cube for the quad version of the equation, or in a 27-point cube for the 9-point formulation. For the Boussinesq equation the former was explicitly done in [22] and the latter was numerically verified in [24,Section 5.7]. The multidimensional consistency also follows from the construction of the GD hierarchy using the direct linearization method [19]. In this construction, lattice shifts are identified with Bcklund transformations, hence the permutability property implies multidimensional consistency.

Continuum limit of the lattice Boussinesq equation (GD3)
We consider the lattice Boussinesq equation in its 9-point scalar form, i.e. Equation (22) for N = 3, For this equation we use the Miwa correspondence (4) with c = −3. As always, we perform a double series expansion of the lattice equation, The first column of coefficients of this expansion is Using this equation we get v 23 = 0 from F 20 = 0. We take the liberty of integrating this without a constant and take v 3 = 0 as the second equation in the hierarchy. We can proceed iteratively and at each step integrate with respect to t 2 to find the hierarchy The first equation of the hierarchy is the potential Boussinesq equation. We observe that every third equation is trivial, v 3k = 0, as expected from the construction of the hierarchy using pseudodifferential operators [6]. Note that in this hierarchy all equations except the first are evolutionary.
The discrete Lagrangian is Unfortunately, not all alien derivatives can be eliminated. Since we do not have any equation in the hierarchy to eliminate first derivatives with respect to t 2 , the derivatives v 2 , v 12 , . . . remain in place. All other derivatives are eliminated from the coefficients where they are alien.
We have omitted the integration constant because higher order terms in the expansion force it to be zero. Equation (27) allows us the eliminate either v or w from the continuous system. Hence we lose one of the components in the continuum limit. It is convenient to make a change of variables in the discrete system, Then in the continuum limit the variable y can be eliminated by Equation (27), which in the new variables reads y = u 2 + u 1 .
(One recognizes the Miura transformation [15].) For the remaining two variables we find the hierarchy where, presumably, every fourth pair of equations is trivial. The construction of the continuous GD4 hierarchy using pseudodifferential operators [6] yields the system φ 2 = 2χ 1 − 2φ 11 as leading order equations. They are related to the leading order equations of the hierarchy (29) by the transformation φ = 4u 1 , χ = 2x 1 + 4u 11 .
In pseudodifferential approach there is no obvious reason to eliminate one of the three variables. The reduction to two variables is forced upon us by the continuum limit, because there does not seem to be any way of performing the limit without getting an ODE relation between the variables, as in Equation (27).

Pluri-Lagrangian structure
For N = 4 we have the Lagrangian L(U, V, W, U 1 , V 1 , W 1 , U 2 , V 2 , W 2 , U 12 , V 12 , W 12 , λ 1 , λ 2 ) As before, we make the change of variables In order to achieve the necessary leading order cancellation, we add to the Lagrangian the discrete exterior derivative of the following discrete 1-form: This 1-form was found by trial and error. It would be interesting to establish a general strategy to find a Lagrangian within a given equivalence class that provides the required cancellation. Without such a strategy, it seems infeasible to apply the continuum limit to higher members of the Gelfand-Dickey hierarchy.
The first few coefficients of the pluri-Lagrangian 2-form are given in Table 7. Note that the Lagrangian 2-form depends on all three fields, u, x, and y. The multi-time Euler-Lagrange equations are equivalent to Equations (28) and (29): they contain both the constraint on y and the hierarchy in u and x.
As with the Boussinesq hierarchy, the elimination of alien derivatives needs a comment. We do not have any equation in the hierarchy to eliminate the derivatives x 2 , x 12 , . . ., so these have to be tolerated in the coefficient L 13 . All other alien derivatives, in particular those of u and y, are eliminated as usual.

Conclusions
The continuum limit procedure developed in [23] can be applied to many discrete pluri-Lagrangian systems. Starting from equations from the ABS list and from discrete Gelfand-Dickey equations, we obtained pluri-Lagrangian structures for among others the Krichever-Novikov, Korteweg-de Vries, and Gelfand-Dickey hierarchies. For many of these hierarchies this is the first time a pluri-Lagrangian structure has been given. The success rate of our method suggests that it is a useful tool to study connections between discrete and continuous integrable systems. Nevertheless, there are lattice systems on which the continuum limit procedure does not seem to apply, an issue which deserves additional investigation. Previously, only a few examples of continuous pluri-Lagrangian 2-form systems were known. The multitude of new examples given in this work contributes to the evidence that a pluri-Lagrangian structure is a relevant attribute of integrability.