Continuum limits of pluri-Lagrangian systems

A pluri-Lagrangian (or Lagrangian multiform) structure is an attribute of integrability that has mainly been studied in the context of multidimensionally consistent lattice equations. It unifies multidimensional consistency with the variational character of the equations. An analogous continuous structure exists for integrable hierarchies of differential equations. We present a continuum limit procedure for pluri-Lagrangian systems. In this procedure the lattice parameters are interpreted as Miwa variables, describing a particular embedding in continuous multi-time of the mesh on which the discrete system lives. Then we seek differential equations whose solutions interpolate the embedded discrete solutions. The continuous systems found this way are hierarchies of differential equations. We show that this continuum limit can also be applied to the corresponding pluri-Lagrangian structures. We apply our method to the discrete Toda lattice and to equations H1 and Q1$_{\delta = 0}$ from the ABS list.


Introduction
A cornerstone of the theory of integrable systems is the idea that integrable equations come in families of compatible equations. In the continuous case these are hierarchies of differential equations with commuting flows. In the discrete case, in particular in the context of equations on quadrilateral graphs (quad equations) this property is known as multidimensional consistency. A classification of multidimensionally consistent quad equations was found by Adler, Bobenko, and Suris [2] and is ofter referred to as the ABS list. Additionally, many integrable equations can be derived from a variational principle. The Lagrangian multiform or pluri-Lagrangian formalism, which grew out of a beautiful insight by Lobb and Nijhoff [14], combines these two aspects of integrability.
The discrete version of the pluri-Lagrangian theory is more developed than the continuous one, and arguably more fundamental. Hence, connecting both sides could lead to a better understanding of the continuous theory. It is known that the lattice parameters of a discrete pluri-Lagrangian system may play the role of independent variables in a corresponding continuous pluri-Lagrangian system of non-autonomous differential equations, see e.g. [14,28]. This paper presents a different connection between discrete and continuous pluri-Lagrangian systems, where the continuous variables interpolate the discrete ones. The lattice parameters describe the size and shape of the mesh on which the discrete system lives, and thus they disappear in the continuum limit. The continuous systems found this way are hierarchies of autonomous differential equations. Pluri-Lagrangian structures for such hierarchies were studied independently of the discrete case in [25].
Some similar continuum limits can be found in the literature, for example in [16,17,18,29] and in particular in [27], where the lattice potential KdV equation is shown to produce the potential KdV hierarchy in a suitable limit. On the level of the pluri-Lagrangian structure, the problem is essentially that of interpolation of discrete variational systems by continuous Lagrangian systems. This was studied in [26] because of its relevance in numerical analysis, in particular for backward error analysis of variational integrators. We will build on the ideas from that work to construct pluri-Lagrangian structures for hierarchies of differential equations that appear as continuum limits of lattice equations. Section 2 contains a crash course on discrete and continuous pluri-Lagrangian systems. Section 3 provides an introduction to Miwa variables, which turn out to be a powerful tool for taking continuum limits. In fact on the level of equations, this is the only tool required to obtain a continuous system. However, that leaves the question whether the resulting differential equations are integrable. In Section 4 we look at the Lagrangian side of the continuum limit; first we review the method of [26] to take continuum limits of classical Lagrangian systems (regardless of their integrability), then we extend these ideas to pluri-Lagrangian systems. By recovering the pluri-Lagrangian structure on the continuous side, the question about integrability of the limit is settled in the affirmative. In Section 5 we study several examples in detail.

Pluri-Lagrangian systems 2.1. Discrete pluri-Lagrangian systems
Consider the lattice Z N with basis vectors e 1 , . . . , e N . To each lattice direction we associate a parameter λ i ∈ C. The equations we are interested in involve the values of a field U : Z N → C on elementary squares in this lattice, or more generally, on d-dimensional plaquettes. Such a plaquette is a 2 d -tuple of lattice points that form an elementary hypercube. We denote it by i 1 ,...,i d (n) = n + ε 1 e i 1 + . . . + ε d e i d ε k ∈ {0, 1} ⊂ Z N , where n = (n 1 , . . . , n N ). Plaquettes are considered to be oriented; an odd permutation of the directions i 1 , . . . , i d reverses the orientation of the plaquette. We will write U ( i 1 ,...,i d (n)) for the 2 d -tuple U ( i 1 ,...,i d (n)) = U (n), U (n + e i 1 ), U (n + e i 2 ), . . . , U (n + e i 1 . . . + e i d ) .
Occasionally we will also consider the corresponding "filled-in" hypercubes in R N , for any permutation σ of i 1 , . . . , i d . Consider a discrete d-surface Γ = { α } in the lattice, i.e. a set of d-dimensional plaquettes, such that the union of the corresponding filled-in plaquettes α α is an oriented topological d-manifold (possibly with boundary). The action over Γ is given by The field U is a solution to the 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 d-surfaces Γ simultaneously.
For d = 1 we have The Euler-Lagrange equations at general elementary corners, are sufficient conditions for U to be a solution to the pluri-Lagrangian problem. For d = 2 we have Since every surface can be constructed out of corners of cubes, it is sufficient to determine the Euler-Lagrange equations on these elementary building blocks. They are These are sufficient conditions for U to be a solution to the pluri-Lagrangian problem. Often, L can be written in a three-leg form which renders the first and last corner equations trivial. In particular, this is the case for all equations from the ABS list. For more details, we refer to [14], [7], [12,Chapter 12], and the references therein.

Continuous pluri-Lagrangian systems
In the continuous case, the the lattice is replaced by a space R N , which we refer to as multi-time. The Lagrangian in this context is a differential d-form where the square brackets denote dependence on the field u : R N → C and an arbitrary number of its partial derivatives. We will always use lower case letters to denote continuous fields, as opposed to the upper case letters used for discrete fields. The field u solves the pluri-Lagrangian problem if for any d-dimensional submanifold Γ of R N it is a critical point of the action S Γ = Γ L with respect to variations that are zero on the boundary of Γ. The multi-time Euler-Lagrange equations, which characterize solutions to the pluri-Lagrangian problem, were derived in [25]. The main idea of that derivation is to approximate any given smooth d-surface by a stepped surface, a piecewise flat surface, the pieces of which are shifted sections of coordinate planes. Analogous to the discrete case, it is sufficient to look at the elementary building blocks of stepped surfaces.
In order to state the multi-time Euler-Lagrange equations we introduce a multi-index notation for partial derivatives. An N -index I is a N -tuple of non-negative integers. There is a natural bijection between N -indices and partial derivatives of u : R N → C. We denote by u I the mixed partial derivative of u, where the number of derivatives with respect to each t i is given by the entries of I. Note that if I = (0, . . . , 0), then u I = u.
In this notation, we will also make use of exponents to compactify the expressions, for example t 3 2 = t 2 t 2 t 2 = (0, 3, 0, . . . , 0). The notation It j should be interpreted as concatenation in the string representation, hence it denotes the multi-index obtained from I by increasing the j-th entry by one. Finally, if the j-th entry of I is non-zero we say that I contains t j , and write I ∋ t j .
For d = 1 the multi-time Euler-Lagrange equations are where δ i δu I denotes a variational derivative in the t i -direction, is obtained from the straight parts of the stepped curve, Equation (1b) from the corners.
For d = 2 the multi-time Euler-Lagrange equations are where Equation (2a) is obtained from the flat pieces of the stepped surface, Equation (2b) from the edges, and Equation (2c) from the corners.
Note that there is no analogue of the lattice parameters in the continuous pluri-Lagrangian framework, but of course it is possible to consider parameter-dependent Lagrangians in the continuous case as well. One way of connecting the discrete and continuous cases is to consider the lattice parameters as independent variables of the continuous system and the discrete independent variables as parameters in the continuous system. This leads to a parameter-dependent non-autonomous PDE, known as the generating PDE, which is discussed for example in [14], [21] and [28]. We will briefly come back to it at the end of this paper.
The main goal of this work is to present a continuum limit procedure for pluri-Lagrangian systems. Instead of switching the roles of parameters and independent variables, we assume that the discrete system lives on a mesh embedded in R N , which is described by the lattice parameters. We then seek a continuous system which interpolates the lattice system.

Miwa variables
To motivate our approach to the continuum limit, we start by considering the opposite direction. 1 The problem of integrable discretization has been studied at impressive length in the monograph [24]. Let us briefly summarize the "recipe" for discretizing Toda-type systems from Section 2.9 of that work. It starts from an integrable ODE with a Lax representation of the form in a Lie algebra g = g + ⊕ g − , where π + denotes projection onto g + . Here L denotes the Lax operator and is not to be confused with a Lagrangian. Such an equation is part of an integrable hierarchy, given by A related integrable difference equation can be formulated in the corresponding Lie group G, with subgroups G + and G − having Lie algebras g + and g − respectively. Any element x ∈ G close to the unit Id ∈ G can be factorized as The difference equation is given by where the tilde · denotes a discrete time step and F (L) = Id + λf (L) for some small parameter λ. Solutions of the differential equation (3) are given by A simultaneous solution to the whole hierarchy (4) takes the form A solution of the discretization (5) is given by Comparing equations (6) and (7), it is natural to identify a discrete step n → n + 1 with a time shift This gives us a map from the discrete space Z N (n 1 , . . . , n N ) into the continuous multi-time R N (t 1 , . . . , t N ). 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 [15]. In the present work we will call the n j discrete coordinates, the λ j lattice parameters and the t i continuous coordinates or times. We will call Equation (8) the Miwa correspondence. Let λ = (λ 1 , . . . , λ N ) and consider the N × N matrix .
Then we can write the Miwa correspondence as where t = (t 1 , . . . , t N ) T , n = (n 1 , . . . , n N ) T , and τ = (τ 1 , . . . , τ N ) T . In other words, we consider the mesh Z N under the affine transformation We will use the Miwa correspondence (8) even if the discrete system is not generated by the recipe described above. In many cases one can justify this in a similar way by considering plane wave factors, solutions of the linearized system. For more on this perspective, see e.g. [19,20,27] and [12,Chapter 5].
For a completely different motivation for Miwa variables, note that for N distinct parameter values λ 1 , . . . , λ N the corresponding vectors are linearly independent. Up to projective transformations, ν is the only curve with that property. It is known as the rational normal curve [11].
To perform the continuum limit of a difference equation involving U : Z N → C, we associate to it a function u : R N → C that interpolates it: 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. Examples can be found in Section 5. 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.

Modified Lagrangians in the classical variational problem
In [26] we performed a continuum limit on Lagrangian systems in the context of variational integrators for ODEs. Given a discrete Lagrangian, we constructed a continuous modified Lagrangian whose critical curves interpolate solutions of the discrete problem. A similar approach can be used in the context of pluri-Lagrangian systems, but first we present the relevant ideas in the context of the classical variational formulation of a P∆E. Here we use parameters h j representing the mesh size of the lattice. In Section 4.2 we will consider the pluri-Lagrangian problem and reinterpret the parameters as Miwa variables.
In the classical discrete variational principle we consider elementary plaquettes of full dimension, so it is sufficient to label them only by position, leaving out the subscripts denoting the direction. We consider Lagrangians L disc ( (n), h 1 , . . . , h d ) depending on the values of the field U : Z d → C on a plaquette (n) and on the mesh sizes h 1 , . . . , h d . As before, we denote lattice shifts by subscripts: We identify points of a discrete solution with mesh size (h 1 , . . . , h d ) with evaluations of an interpolating field u : Using a Taylor expansion we can write the discrete Lagrangian L disc ( (n), h 1 , . . . , h d ) as a function of the interpolating field u and its derivatives, where the square brackets denote dependence on u and any number of its partial derivatives. So far we have only written the discrete Lagrangian as a function of the continuous field. The corresponding action is still a sum: We want to write the action as an integral. This can be done using the Euler-Maclaurin formula, which relates sums to integrals [1, Eq. 23.1.30]: where B i denote the Bernoulli numbers 1, − 1 2 , 1 6 , 0, − 1 30 , 0, · · · . Applying this to L disc in each of the lattice directions, we obtain the meshed modified Lagrangian The power series in the Euler-Maclaurin Formula generally does not converge. The same is true for the series defining L mesh . Formally, it satisfies This property also holds locally, The word meshed refers to the fact that the discrete system provides additional structure for the continuous variational problem. In the meshed variational problem, nondifferentiable fields are admissible as long as their singular points are consistent with the mesh, i.e. if they only occur on the boundaries of mesh cells. This imposes additional conditions on critical curves, related to the natural boundary conditions and to the Weierstrass-Erdmann corner conditions (see e.g. [10, Sec. 6 and 13] for these two concepts). In [26] these conditions were used to turn the meshed modified Lagrangian into a true modified Lagrangian which does not depend on higher derivatives. We will not discuss this method here. Instead we will find that the pluri-Lagrangian structure provides us with simpler tools to eliminate unwanted derivatives.
Because the power series defining L mesh usually does not converge, we introduce the following concept of criticality.
In the discrete case the definition is analogous, with integrals replaced by sums.
Note that in contrast to [26] we do not consider parameter-dependent families of fields. This is because we do not want the lattice parameters to survive in the continuum limit; u should not depend on the lattice parameters. A welcome consequence of this restriction is that it allows us to avoid much of the cumbersome analysis of [26]. In the current setting the following property is quite obvious.
if and only if it satisfies the Euler-Lagrange equations with a defect of order O(h k+1

From discrete to continuous pluri-Lagrangian structures
In the pluri-Lagrangian context we consider a discrete Lagrangian d-form in a higher dimensional lattice Z N , N > d. Furthermore, from now on the lattice parameters are interpreted as Miwa variables, hence they will not have the immediate interpretation of mesh size. Through the Miwa correspondence (8) they still determine a lattice embedded in the continuous space R N , albeit a very skew one. 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 calculate the modified Lagrangian in the transformed coordinate system. The eval- . of a continuous field correspond to a discrete field evaluated on the plaquette located at where now the differential operators correspond to the lattice directions under the Miwa correspondence, and the parameters have been absorbed into the differential operators: The meshed modified Lagrangian in Miwa coordinates is given by Note that L Miwa depends on a field parameterized in Miwa variables, whereas L disc and L mesh are inherited form the previous subsection and thus depend on a field described in orthonormal coordinates.
Lemma 3. Consider a filled-in plaquette of the embedded lattice, A c,λ,τ ( i 1 ,...,i d (n)), and let η k be the 1-forms dual to the Miwa shifts, Proof. In Equation (9) we have the corresponding result for L mesh , so the proof is a simple change of variables: We want to use this result for plaquettes in arbitrary directions. This suggests the Lagrangian d-form Up to a truncation error, this d-form can be written in a much more convenient way. Let T N denote truncation of a power series after degree N in each variable, Lemma 4. Assume that every term in the power series L Miwa is of strictly positive degree in each λ i , then Note in Equation (10) that the factors (−1) Proof of Lemma 4. First observe that, just like the discrete Lagrangian, the Lagrangian L Miwa ([u], λ i 1 , . . . , λ i d ) is skew-symmetric as a function of (λ i 1 , . . . , λ i d ). Therefore, the coefficients L i 1 ,...,i d [u] are skew-symmetric as a function of (i 1 , . . . , i d ).
We pair the form , this can be written as Since the first sum is over all d-tuples (i 1 , . . . , i d ) with strictly positive integer entries, permuting (i 1 , . . . , i d ) yields a different term of this sum. Hence the additional summation over permutations σ ∈ S d amounts to multiplication by d!. We find Theorem 5. Let L disc be a discrete Lagrangian d-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 (10). Consider the differential d-form built out of the coefficients of L Miwa . Then a field u : R N → C is a solution to the continuous pluri-Lagrangian problem for L if and only if the corresponding discrete fields are N -critical for the discrete pluri-Lagrangian problem for L disc .
Proof. Consider the (d + 1)-dimensional cube in the lattice Z N . It corresponds to a (d + 1)-dimensional parallelotope in R N . Combining Lemma 3 and Lemma 4 we find that Note that the integral ∂Pτ L[u] still depends on the λ i because the parallelotope P τ depends on them. From this relation it follows that if u is a solution to the pluri-Lagrangian problem for L, then the discrete action is N -critical.
On the other hand, if the discrete fields U τ are N -critical for the pluri-Lagrangian problem for every τ ∈ R N , then the continuous action for the d-form L[u] is N -critical on every paralellotope P τ . Therefore, the continuous action is N -critical on any corner of such a paralellotope (see Figure 3). In [25]

Eliminating alien derivatives
Suppose a pluri-Lagrangian d-form in R N produces multi-time Euler-Lagrange equations of evolutionary type, for k ∈ {d, d + 1, . . . , N }.
A multi-index I is said to be native or alien if the corresponding derivative u I is of that type.
We would like the coefficient L i 1 ,...,i d to be {i 1 , . . . , i d }-native. A naive approach would be to use the multi-time Euler-Lagrange equations (11) to eliminate all alien derivatives. Let R i 1 ,...,i d denote the operator that replaces all {i 1 , . . . , i d }-alien derivatives using (11). We denote the native version of the pluri-Lagrangian coefficients by L i 1 ,...,i d = R i 1 ,...,i d (L i 1 ,...,i d ) and the d-form with these coefficients by L. A priori there is no reason to believe that the d-form L will be equivalent to the original pluri-Lagrangian d-form L. For example, the 1-dimensional Lagrangian L(u, u t , u tt ) = 1 2 uu tt leads to the Euler-Lagrange equation u tt = 0, but any curve is critical for the Lagrangian L(u, u t , u tt ) = 0. However, in many cases the pluri-Lagrangian structure guarantees that L and L have the same critical fields. The condition for d = 2 might seem restrictive, but given a Lagrangian 2-form, we can often find an equivalent one with coefficients L 1j [u] that satisfy this condition by inspection.
Proof of Theorem 7. In this proof we consider the variation operator δ as the vertical exterior derivative in the variational bicomplex. A short introduction to the variational bicomplex is given in Appendix A.
First we consider the case d = 1. Let Hence on solutions of the pluri-Lagrangian problem for L there holds that Using the assumption that no alien derivatives occur in L 1 , we can simplify this to The fact that δL is exact with respect to d implies that δ Γ L = 0 for all curves Γ and all variations that are zero on the endpoints of Γ. Hence u is a solution to the pluri-Lagrangian problem for L. Now we consider the case d = 2. Let Note that R ij commutes with both D t i and D t j . We have On solutions of the pluri-Lagrangian problem for L there holds that where we have left out the R ij because the L 1j do not contain any alien derivatives. For the same reason, only terms where J is {i, j}-native can be nonzero, so in all nonvanishing terms we find F ij,J = u J . Therefore, This implies that δ Γ L = 0 for all surfaces Γ and all variations that are zero on the boundary of Γ. Hence u is a solution to the pluri-Lagrangian problem for L.

Examples
The plan for this section is as follows. We begin with the 1-form case and discuss the continuum limit of the discrete Toda lattice. After that we present three examples for the 2-form case. The first one is a linear quad equation. This will help us understand how to proceed for the two non-linear quad equations that follow, H1 and Q1 δ=0 from the ABS list. In each of the examples we first perform the continuum limit on the level of equations and then discuss the pluri-Lagrangian structure.

Equation
The Toda lattice consists of a number of particles on a line with an exponential nearestneighbor force. If we denote the positions of the particles with then their motion is described by the equation There are two common conventions regarding boundary conditions: periodic (formally q [N+1] ≡ q [1] ) and open-end (formally q [0] ≡ +∞ and q [N+1] ≡ −∞). An integrable discretization of the Toda lattice is given by [24, Chapter 5] where the subscripts i and −i denote forward and backward shifts respectively and λ i is a lattice parameter. We use the Miwa correspondence (8) with c = 1 to identify discrete steps with continuous time shifts We plug these identifications into Equation (12) and perform a Taylor expansion in λ: where the subscripts i are a shorthand for t i and denote partial derivatives. As long as one remembers that discrete fields are printed in upper case and continuous fields in lower case, there should be no confusion between partial derivatives and lattice shifts. In the leading order term we recognize the first Toda equation Using this equation, we find that the coefficient of Under the differentiation one can recognize the second Toda equation Similarly, the higher order terms correspond to the subsequent equations of the Toda hierarchy.

Pluri-Lagrangian structure
A pluri-Lagrangian structure for the discrete Toda equation was studied in [6]. The Lagrangian is given by Performing a Taylor expansion and applying the Euler-Maclaurin formula as in Section 4.2, we obtain with coefficients . . . By Theorem 5, these are the coefficients of a pluri-Lagrangian 1-form L = i L i dt i for the Toda hierarchy (13), (14), · · · . Note that L 3 contains derivatives with respect to t 2 . We replace these using the second Toda equation and find Similarly one can obtain L i for i ≥ 4. By Theorem 7, the corresponding 1-form L is equivalent to L. The Lagrangian 2-form L is identical to the one that was found in [22] using the variational symmetries of the Toda lattice.

Equation
Consider the linear quad equation It is a discrete analogue of the Cauchy-Riemann equations [5] and also the linearization of the lattice potential KdV equation, which will be discussed in Section 5.3. Therefore all the results in this section are consequences of those in Section 5.3. Nevertheless, this simple quad equation is a good subject to illustrate some of the subtleties of the continuum limit procedure.
To get meaningful equations in the continuum limit, we need to write the quad equation in a suitable form. Since in the Miwa correspondence the parameter enters linearly in the t 1 -coordinate and with higher powers in the other coordinates, the leading order of the expansion of the shifts of U will only contain derivatives with respect to t 1 . Other derivatives enter at higher orders. Since we want to obtain PDEs in the continuum limit, not ODEs, we must require that the leading order of the expansion yields a trivial equation.
Written in terms of difference quotients, Equation (16) reads . .), etc., this would yield u t 1 = −u t 1 in the leading order of the expansion. In order to avoid this, we introduce new parameters or, equivalently, Inside the brackets we find u t 1 = u t 1 in the leading order if we set U = u(t 1 , . . .), U i = u(t 1 + λ i , . . .), etc., which is trivial as desired.
We use the Miwa correspondence (8) with c = −2. This choice will give us a nice normalization of the resulting differential equations. We apply the Miwa correspondence to Equation (17) and expand to find a double power series in λ 1 and λ 2 , where F ji = −F ij . The factor (−1) i+j 4 ij is chosen to normalize the F 0j , but does not influence the final result. The first few of these coefficients are We see that the flows corresponding to even times are trivial. In the odd orders we find a hierarchy of linear equations, For i ≥ 1, the equations F ij = 0 are consequences of these equations.

Pluri-Lagrangian structure
The linear quad equation (16) possesses a pluri-Lagrangian structure [5,13], The following Lemma will help us put this Lagrangian in a more convenient form.
is a null Lagrangian (i.e. its multi-time Euler-Lagrange equations are trivially satisfied) Proof. Consider the discrete 1-form given by η(U, U i ) = U U i and η(U i , U ) = −U U i . Its discrete exterior derivative is Just like in the continuous case, this means that the action of L 0 over any discrete surface only depends on values of U at the boundary of the surface. Hence all fields are critical with respect to variations in the interior.
Using Lemma 8, we see that the Lagrangian (18) is equivalent to (denoted with = by abuse of notation) or, in terms of the parameters λ k , Since the Taylor expansion of (U i − U j ) 2 contains a factor λ i − λ j , the expansion of the Lagrangian does not contain any negative order terms. In fact all zeroth order terms vanish as well, so Theorem 5 applies: the coefficients of the power series We find . . .
We will not study this example in more detail. Instead we move on to one of its non-linear cousins.

Equation
Consider equation H1 from the ABS list [2], also known as the lattice potential Korteweg-de Vries (lpKdV) equation, We would like write Equation (19) in terms of difference quotients. To achieve this, we identify α 1 = −λ −2 1 and α 2 = −λ −2 2 . Then Equation (19) is equivalent to The left hand side is now a product of meaningful difference quotients, but the right hand side explodes as the parameters tend to zero. (Setting α i = −λ 2 i instead would cause the same problem as in the first attempt of Section 5.2.) To avoid this we make a nonautonomous change of variables V (n 1 , . . . , n N ) = U (n 1 , . . . , n N ) + Then the lpKdV equation takes the form This is the form in which the lpKdV equation was originally found and studied, usually with parameters p = λ −1 1 and q = λ −1 2 , see [19] for an overview. In terms of difference quotients, the equation reads If we identify U = u(t 1 , . . .), U i = u(t 1 + λ i , . . .), etc., then the negative powers of the parameters cancel. In the leading we find the tautological equation u t 1 −u t 1 = 0. Therefore, this form of the difference equation is a suitable candidate for the continuum limit. Again we use the Miwa correspondence (8) with c = −2. From Equation (20) we find a double power series in λ 1 and λ 2 , where F ji = −F ij . The first few of these coefficients are where once again we use the subscript i rather than t i to denote partial derivatives of u. We see that the flows corresponding to even times are trivial. In the odd orders we find the pKdV equations, For i ≥ 1, the equations F ij = 0 are consequences of these equations.
A pluri-Lagrangian description of Equation (19) was found in [14], the Lagrange function itself goes back to [8]. It reads Using Lemma 8, we see that this Lagrangian is equivalent to (denoted with = by abuse of notation) In terms of U and λ it is (up to a constant) Lemma 9 implies that L is equivalent to To see why this Lagrangian is preferable, do a first order Taylor expansion of the logarithm and admire the cancellation. Thanks to this cancellation we avoid terms of non-positive order in the series expansion. Applying the Miwa correspondence (8) with c = −2, a Taylor expansion, and the Euler-Maclaurin formula to the Lagrangian (21), we obtain a power series whose coefficients define a continuous pluri-Lagrangian 2-form for the KdV hierarchy. The first row of coefficients reads: Now that we have disposed of the alien derivatives in the L 1j , we can use Theorem 7 to eliminate the remaining alien derivatives in all other L ij . For i < j ≤ 5, the coefficients obtained this way are displayed in Table 1.
On the level of equations we could have restricted to the odd-numbered coordinates t 1 , t 3 , . . . from the beginning. However, on the level of Lagrangians we need to consider the evennumbered coordinates as well, at least in the theoretical arguments, because otherwise there is no interpretation for the (generally non-zero) coefficients of λ 2i 1 λ 2j 2 in the power series L Miwa

The double continuum limit of Wiersma and Capel
In [27] Wiersma and Capel presented a continuum limit of the lpKdV equation which is equivalent to equation (20) under the transformation µ i = λ −1 i . Their procedure consists of two steps. First they obtain a hierarchy of differential-difference equations. A second continuum limit, applied to any single equation of this hierarchy, then yields the potential KdV hierarchy. Some ideas concerning this limit procedure were already developed in [20,23]. Here we will summarize both limits in one step.
Consider an interpolating function u. If V (n, m) = U (n − m, m) = u(t 1 , t 3 , t 5 , . . .), then after the double limit of [27], lattice shifts correspond to multi-time shifts as follows: It is also known as the cross-ratio equation [4,19] and as the lattice Schwarzian KdV equation [12,Chapter 3]. As before, we would like to view Equation (24) as a consistent numerical discretization of some differential equation. To achieve this, we identify α 1 = λ 2 1 and α 2 = λ 2 2 . Then Equation (19) is equivalent to If we identify V = v(t 1 , . . .), V i = v(t 1 + λ i , . . .), etc., then the leading order expansion yields v 2 t 1 − v 2 t 1 = 0. This is a tautological equation, as desired. Hence in this case there is no need for an additional change of variables.
Once more we use the the Miwa correspondence (8) with c = −2. A Taylor expansion of (25) yields . . .
We assume that v 1 = 0. Then we see that the flows corresponding to even times are trivial.
In the odd orders we find the hierarchy of Schwarzian KdV equations, For i ≥ 1, the equations F ij = 0 are differential consequences of these equations.

Pluri-Lagrangian structure
A Pluri-Lagrangian description of Equation (24) was found in [14] which is equivalent to Each term of the series L Miwa constructed form this discrete Lagrangian contains strictly positive powers of both λ i and λ j . Thus by Theorem 5 we can identify the coefficients of this power series with the coefficients of a pluri-Lagrangian 2-form. Some of these coefficients, after eliminating alien derivatives, are given in Table 2.
Again we can restrict the pluri-Lagrangian formulation to a space of half the dimension and consider as a pluri Lagrangian 2-form for the non-trivial equations of the SKdV hierarchy.

The generating PDE of Nijhoff, Hone, and Joshi
Nijhoff, Hone, and Joshi [21] introduced a non-autonomous PDE for a function z n,m (t, s) depending on a pair of continuous variables (s, t), and a pair of parameters (m, n). They noted that the flow of this PDE in continuous (s, t)-coordinates commutes with the difference equations (z n,m − z n+1,m )(z n,m+1 − z n+1,m+1 ) (z n,m − z n,m+1 )(z n+1,m − z n+1,m+1 ) = s t .
Equation (28) is nothing but equation Q1 δ=0 . Hence it is possible to switch between the continuous and discrete picture by reversing the roles of parameters and independent variables. The main feature of the PDE in question is that it generates the SKdV hierarchy 2 through the identification z n,m (t, s) = v x 1 + 2n 3s 3 2 , . . . , , . . . .  Generating PDE for z n,m (t, s) Quad equation (28) Hierarchy (26) role reversal (n, m) ↔ (t, s) Miwa expansion continuum limit Figure 4: The continuum limit is compatible with the relations found in [21] Because of this it has become known as the generating PDE [14,28] for the SKdV hierarchy.
Hence our continuum limit of Q1 δ=0 is implicitly present in [21]. The relation between the (non-autonomous) generating PDE, the quad equation, and the hierarchy of (autonomous) PDEs is illustrated in Figure 4.

Conclusion
We have presented a method to perform continuum limits of discrete pluri-Lagrangian systems. In this approach, a single (parameter-dependent) discrete equation produces a full hierarchy of differential equations, and the pluri-Lagrangian structure is carried over from the discrete system to the continuous one. Although the method can be stated in a general way, it can only be executed if we can find a form of the discrete equation and its Lagrangian that allows a suitable Taylor expansion in the parameters. Finding such a form is a non-trivial task. We solved this on a case-by-case basis for a few important examples. In a future publication we will discuss many more examples, including all ABS equations of type Q.

A. The variational bicomplex
This is a minimal introduction to the variational bicomplex. Much more on this topic can be found for example in [9,Chapter 19].
Our starting point is the idea that the exterior derivative can be split into a vertical part δ and a horizontal part d. An (a, b)-form is a differential (a + b)-form structured as An important property of classical Lagrangian systems is that changing the Lagrangian by a full derivative (or divergence) does not affect the Euler-Lagrange equations. The following Proposition is the pluri-Lagrangian generalization of this property. Proof. Since the horizontal exterior derivative d anti-commutes with the interior product operator ι pr V for the prolongation of a vertical vector field V , it follows that for any variation V of the field u that is zero on the boundary of a manifold Γ: