1Introduction AVariationalPerspectiveonContinuumLimitsofABSandLatticeGDEquations

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A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations

Mats VERMEEREN

Institut f¨ur Mathematik, MA 7-1, Technische Universit¨at Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany

E-mail: vermeeren@math.tu-berlin.de

URL: http://page.math.tu-berlin.de/~vermeer/

Received November 20, 2018, in final form May 16, 2019; Published online June 03, 2019 https://doi.org/10.3842/SIGMA.2019.044

Abstract. 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.

Key words: continuum limits; pluri-Lagrangian systems; Lagrangian multiforms; multidimensional consistency

2010 Mathematics Subject Classification: 37K10; 39A14

1 Introduction

This paper continues the work started in [25], where we established a continuum limit procedure for lattice equations with apluri-Lagrangian (also calledLagrangian 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 [25]. In Sections 3and 4 we present examples form the ABS list [2]. 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 [22]. The code is available at https://github.com/mvermeeren/pluri-lagrangian-clim.

1.1 The discrete pluri-Lagrangian principle

Most of the lattice equations we will consider are quad equations, difference equations of the form

Q(U, U₁, U₂, U₁₂, α₁, α₂) = 0,

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where subscripts of the field U:Z²→Cdenote lattice shifts,

U ≡U(m, n), U1 ≡U(m+ 1, n), U2≡U(m, n+ 1), U12≡U(m+ 1, n+ 1), and α_i∈Care parameters associated to the lattice directions.

Even though the equations live inZ², we require that we can consistently implement them on every square in a higher-dimensional latticeZ^N. This property ofmultidimensional consistency is an important attribute of integrability for lattice equations, see for example [2, 5], or [9].

A necessary and sufficient condition for multidimensional consistency is that the equation is consistent around the cube:

U1

U13

U3

U

U12

U123

U23

U₂

α1

α2

α₃

Figure 1. A quad equation is consistent around the cube ifU123can be uniquely determined fromU,U1, U2 andU3. If in additionU123 is independent ofU, then the equation satisfies the tetrahedron property.

Definition 1.1. Given lattice parameters α₁, α₂, and α₃ and field values U, U₁, U₂, and U₃, we can use the equations

Q(U, U_i, U_j, U_ij, λ_i, λ_j) = 0, 1≤i < j ≤3

to determine U₁₂,U₁₃, and U₂₃. Then we can use each of the three equations Q(Ui, Uij, Uik, Uijk, λj, λk) = 0, (i, j, k) even permutation of (1,2,3)

to determine U₁₂₃. If these three values agree for all initial conditionsU,U₁,U₂, andU₃ and all parameters α1,α2, and α3, then the equation isconsistent around the cube.

Multidimensionally consistent equations on quadrilateral graphs, satisfying certain additional assumptions, were classified by Adler, Bobenko and Suris in [2]. The list of equations they found is widely known as the 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 e₁, . . . ,e_N. To each lattice direction we associate a parameter λi ∈ C. We denote an elementary square (aquad) in the lattice by

i,j(n) =

{

n+ε1e_i+ε2e_j

|

ε1, ε2 ∈ {0,1}

}

⊂Z^N,

wheren= (n₁, . . . , n_N). Quads are considered to be oriented; interchanging the indicesiand j reverses the orientation. We will write U(i,j(n)) for the quadruple

U(i,j(n)) =

(

^U^{(n), U}⁽ⁿ⁺^ei), 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+µ₁e_i+µ₂e_j

|

µ₁, µ₂ ∈[0,1]

}

^⊂R^N,

on which we consider the orientation defined by the basis (ei,e_j) of the tangent space.

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Figure 2. Visualization of a discrete 2-surface inZ³.

The role of a Lagrange function is played by a discrete 2-form L(U(i,j(n)), λi, λj),

which is a function of the values of the field U:Z^N → C on a quad and of the corresponding lattice parameters, satisfyingL(U(j,i(n)), λj, λi) =−L(U(i,j(n)), λi, λj).

Consider a discrete surface Γ ={α} in the lattice, i.e., a set of quads, such that the union of the corresponding filled-in squaresS

αα is an oriented topological 2-manifold (possibly with boundary). The action over Γ is given by

S_Γ= X

i,j(n)∈Γ

L(U(i,j(n)), λi, λj).

Definition 1.2. 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 1.3. 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-formL= ∆η, defined by

L(U, U_i, U_j, U_ij, λ_i, λ_j) =η(U, U_i, λ_i) +η(U_i, U_ij, λ_j)−η(U_j, U_ij, λ_i)−η(U, U_j, λ_j).

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 introduced the pluri-Lagrangian (or Lagrangian multiform) idea, and to the reviews [5] and [9, Chapter 12], and the references therein.

1.2 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 fieldu:R^N →Cand any number of its derivatives.

Definition 1.4. The field u solves the continuous pluri-Lagrangian problem for L if for any embedded surface Γ ⊂ Rⁿ and any variation δu which vanishes near the boundary ∂Γ, there holds

δ Z

Γ

L= 0.

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The first question about continuous pluri-Lagrangian systems is to find a set of equations characterizing criticality in the pluri-Lagrangian sense. If L = P

i<j

L_ij[u] dt_i ∧dt_j, then these equations are

δijL_ij

δu_I = 0 ∀I 63ti, tj, (1.1)

δijL_ij

δu_It_j −δ_ikL_ik

δu_It_k = 0 ∀I 63ti, (1.2)

δijL_ij

δu_It_i_t_j +δjkL_jk

δu_It_j_t_k + δ_kiL_ki

δu_It_k_t_i = 0 ∀I, (1.3)

fori,j, andk distinct, where δ_ij

δuI

=

∞

X

α,β=0

(−1)^α+β d^α dt^α_i

d^β dt^β_j

∂

∂u_Itα it^β_j

,

and uI denotes a partial derivative corresponding to the multi-index I. We write that I 3tk if the entry inI corresponding to t_kis nonzero, i.e., if at least one derivative is taken with respect to t_k, and I 63t_k otherwise. Compared tou_I, the field u_Itα

it^β_j hasα additional derivatives with respect tot_i and β additional derivatives with respect tot_j.

We call equations (1.1)–(1.3) the multi-time Euler–Lagrange equations. They were derived in [23].

2 Continuum limit procedure

In this section we review the essentials of the continuum limit procedure for multidimensionally consistent lattice equations and their pluri-Lagrangian structure, presented in [25]. The continuum limit on the level of the equations has its roots in [16,27]. On the level of the Lagrangian structure, a significant precursor is [28], in which the continuum limit of a Lagrangian 1-form system is presented.

2.1 Miwa variables

We construct a map from the latticeZ^N(n₁, . . . , n_N) to the continuous multi-timeR^N(t₁, . . . , t_N) as follows. We associate a parameterλi with each lattice direction and set

ti = (−1)ⁱ⁺¹

n1

λⁱ₁

i +· · ·+nN

λⁱ_N i

.

Note that a single step in the lattice (changing one nj) affects all the times ti, hence we are dealing with a very skew embedding of the lattice. We will also consider a slightly more general correspondence,

ti = (−1)ⁱ⁺¹

n1cλⁱ₁

i +· · ·+nN

cλⁱ_N i

+τi, (2.1)

for constants c, τ₁, . . . , τ_N describing a scaling and a shift of the lattice. The variablesn_j andλ_j are known in the literature as Miwa variables and have their origin in [16]. We will call equation (2.1) theMiwa correspondence.

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We denote the shift of U in the i-th lattice direction by Ui. If U(n) = u(t₁, . . . , tN), It is given by

U_i =U(n+e_i) =u

t₁+cλ_i, t₂−cλ²_i

2 , . . . , t_n−(−1)^Ncλ^N_i N

,

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 t₁-derivatives in the leading order, because only in the t₁-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 continuumlimit: sendingλi→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 λi, where U is defined on a mesh that is embedded inR^N using the Miwa correspondence. Sinceλi 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.

2.2 Continuum limits of Lagrangian forms

We sketch the limit procedure for pluri-Lagrangian structures introduced in [25]. Consider N pairwise distinct lattice parameters λ₁, . . . , λN and denote by e₁, . . . ,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:

e_i7→v_i=

cλi,−cλ²_i

2 , . . . ,(−1)^N+1cλ^N_i N

fori= 1, . . . , N.

We start from a discrete Lagrangian two-formL_disc. A discrete field can be recovered from a continuous field by evaluating it at u(t), u(t+v_i), u(t+v_i+v_j),· · ·. We define

L_disc([u], λ₁, λ₂) =L_disc

u, u+∂₁u+1

2∂₁²u+· · ·, u+∂₂u+1

2∂₂²u+· · ·, u+∂₁u+∂₂u+1

2∂²₁u+∂₁∂₂u+1

2∂₂²u+· · ·, λ₁, λ₂

,

where the differential operators correspond to the lattice directions under the Miwa correspondence,

∂k=

N

X

j=1

(−1)^j+1cλ^j_k j

d dtj

.

As the subscript indicates,L_discis 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

L_Miwa([u], λ₁, λ₂) =

∞

X

i,j=0

B_iB_j

i!j! ∂₁ⁱ∂^j₂L_disc([u], λ₁, λ₂),

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which is a formal power series inλ₁ and λ₂. Then by construction we have the formal equality L_disc(U(12(n)), λ1, λ2) =

Z

¹²^L^Miwa^{([u], λ}¹^{, λ}²⁾^η¹^∧^η²^,

where (η₁, . . . , η_N) are the 1-forms dual to the Miwa shifts (v₁, . . . ,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

L= X

1≤i<j≤N

L_Miwa([u], λ_i, λ_j)η_i∧η_j.

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 seriesL_Miwa.

Theorem 2.1 ([25]). 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 both λi, i.e., such that L_Miwa is of the form

L_Miwa([u], λ₁, λ₂) =

∞

X

i,j=1

(−1)^i+jc²λⁱ₁ i

λ^j₂

j L_i,j[u].

Then the differential 2-form L= X

1≤i<j≤N

L_i,j[u] dti∧dtj

is a pluri-Lagrangian structure for the continuum limit hierarchy, restricted to R^N.

In all examples presented below it was also verified by direct calculation that solutions to the continuum limit equations are indeed critical fields for the corresponding pluri-Lagrangian structure.

Closely related to our definition of a discrete pluri-Lagrangian system is the property that on solutions

L_disc(U(i,j(n+e_k)), λi, λj)−L_disc(U(i,j(n)), λi, λj)

+L_disc(U(j,k(n+e_i)), λj, λk)−L_disc(U(j,k(n)), λj, λk)

+L_disc(U(k,i(n+e_j)), λk, λi)−L_disc(U(k,i(n)), λk, λi) = 0, (2.2) which implies that the action sum over any closed surface is zero. Some authors argue that this is the fundamental property of the Lagrangian theory of multi-dimensionally consistent equations [9, 13, 14,28]. (The authors that take this perspective mostly use the term “Lagrangian multiform” in place of our “pluri-Lagrangian”.) In the continuum limit, this turns into the property that the action integral vanishes on closed surfaces, when evaluated on solutions. In other words, equation (2.2) implies that the 2-formL found in the continuum limit is closed on solutions of the limit hierarchy.

2.3 Eliminating alien derivatives

Suppose a pluri-Lagrangian 2-form in R^N produces multi-time Euler–Lagrange equations of evolutionary type,

utk =f_k[u] fork∈ {2,3, . . . , N}.

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If this is the case, then the differential consequences of the multi-time Euler–Lagrange equations can be written in a similar form,

uI=fI[u] withI 3t_k for somek∈ {2,3, . . . , N}, (2.3) where the multi-index I in fI is a label, not a partial derivative. In this context it is natural to consider t₁ as a space coordinate and the others as time coordinates. If the multi-time Euler–Lagrange equations are not evolutionary, equation (2.3) still holds for a reduced set of multi-indices I.

Definition 2.2. A mixed partial derivativeu_Iis called{i, j}-native if each individual derivative is taken with respect to ti,tj or the space coordinate t₁, i.e., if

I 3t_k ⇒ k∈ {1, i, j}.

If u_I is not {i, j}-native, i.e., if there is a k 6∈ {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 coefficientL_i,j, we will usenative and alien without mentioning the indices.

We would like the coefficientL_i,j to contain only native derivatives. A naive approach would be to use the multi-time Euler–Lagrange equations (2.3) to eliminate alien derivatives. Let Ri,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 =Ri,j(Li,j) and the 2-form with these coefficients byL. A priori there is no reason to believe that the 2-formLwill be equivalent to the original pluri-Lagrangian 2-formL, but using the multi-time Euler–Lagrange equations one can derive the following result [25].

Theorem 2.3. If for all j the coefficientL_1j does not contain any alien derivatives, then every critical field u for the pluri-Lagrangian 2-formL is also critical for L.

In practice, given a Lagrangian 2-form, we can make it fulfill the conditions on the L_1j 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.

3 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 3.1we present this general strategy. In Sections 3.2–3.7we will discuss each individual equation of type Q.

3.1 Three leg forms and Lagrangians

All quad equations Q(V, V1, V2, V12, λ1, λ2) = 0 from the ABS list have a three leg form:

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

ψ(v, v⁰, λ, µ) =ψ₁(v, λ, µ) +ψ₂(v⁰, λ, µ)

of the continuous variables, from which we can recover Ψ(V, W, λµ) by plugging in suitable approximations to v and v⁰. 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 vt1 is approximated by difference quotients such as ^V⁻¹_2λ^−V

1 and ^V_2λ^−V²

2 . We identify Ψ(V, W, λµ) =ψ

V +W

2 ,V −W 2λ , λ, µ

.

All equations of the Q-list can be written in the form Q(V, V₋₁, V₂, V_−1,2, λ₁, λ₂) =ψ

V +V₋₁

2 ,V −V₋₁ 2λ₁ , λ₁, λ₁

−ψ

V +V₂

2 ,V −V₂ 2λ₂ , λ₂, λ₂

−ψ

V +V_−1,2

2 , V −V_−1,2

2(λ1−λ2), λ₁−λ₂, λ₁+λ₂

. (3.1)

As suggested by the D₄-symmetry of a quad equation, in particular by Q(V, V₋₁, V₂, V_−1,2, λ₁, λ₂) =−Q(V, V₂, V₋₁, V_−1,2, λ₂, λ₁),

we require that

ψ₁(v,−λ, µ) =−ψ₁(v, λ, µ), ψ₂(−v⁰,−λ, µ) =−ψ₂(v⁰, λ, µ).

Furthermore, we require that ψ2(v⁰,−λ, µ) =ψ2(v⁰, λ, µ)

and ψ(v, v⁰,0,0) = 0. As we will see below, all ABS equations of type Q have a three-leg form that satisfies these conditions.

We would expect the first nonzero terms in the series expansion at first order inλ₁,λ₂, but using the symmetry ofψ one can check that

Q(V, V₋₁, V₂, V_−1,2, λ₁, λ₂) =O

(

^λ²₁⁺^λ²₂

)

^.

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 t₁, we get nothing at all.

Equation (3.1) also reveals a reason for considering the three-leg form with a “downward”

diagonal leg, as in Fig.3(b): the difference quotient _2(λ^V^−1,2^−V

2−λ1) can be expanded in a double power series, but its “upward” analogue _2(λ^V^1,2^−V

1+λ2) cannot.

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(a) (b) (c) (d)

Figure 3. The stencils on four adjacent quads for (a) the three-leg form in the usual orientation, (b) the three-leg form after time-reversal, (c) the triangle form for the Lagrangian, and (d) an Euler–Lagrange equation in a planar lattice.

To find a Lagrangian for equation (3.1), we follow [2,4] and integrate the leg functionψ. We take

χ₁(v, λ, µ) = 2 λ

Z

ψ₁(v, λ, µ)dv and χ₂(v⁰, λ, µ) = 2 Z

ψ₂(v⁰, λ, µ)dv⁰. Then

χ1(v,−λ, µ) =χ1(v, λ, µ),

χ₂(−v⁰, λ, µ) =χ₂(v⁰,−λ, µ) =χ₂(v⁰, λ, µ), and χ=χ₁+χ₂ satisfies

λ 2

∂

∂vχ(v, v⁰, λ, µ) +1 2

∂

∂v⁰χ(v, v⁰, λ, µ) =ψ(v, v⁰, λ, µ).

Now

Λ(V, W, λ, µ) =λ χ

V +W

2 ,V −W 2λ , λ, µ

gives the terms of the Lagrangian in triangle form:

L(V, V₁, V₂, λ₁, λ₂) = Λ(V, V₁, λ₁, λ₁)−Λ(V, V₂, λ₂, λ₂)−Λ(V₁, V₂, λ₁−λ₂, λ₁+λ₂)

=λ1χ

V +V1

2 ,V −V1

2λ₁ , λ1, λ1

−λ2χ

V +V2

2 ,V −V2

2λ₂ , λ2, λ2

−(λ₁−λ₂)χ

V₁+V₂

2 , V₁−V₂

2(λ₁−λ₂), λ₁−λ₂, λ₁+λ₂

. (3.2)

Note the symmetries of Λ:

Λ(V, W, λ, µ) = Λ(W, V, λ, µ) =−Λ(V, W,−λ, µ).

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 3.1. Solutions to the quad equation Q = 0 in the plane, with Q given by equation (3.1), are critical fields for the action of the Lagrangian given by equation (3.2).

Proof . We have

∂

∂V L(V, V1, V2, λ1, λ2) =λ1 ∂

∂Vχ

V +V1

2 ,V −V1

2λ₁ , λ1, λ1

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−λ₂ ∂

∂V χ

V +V₂

2 ,V −V₂ 2λ₂ , λ₂, λ₂

= λ₁ 2 χ⁰₁

V +V₁ 2 , λ₁, λ₁

+1

2χ⁰₂

V −V₁ 2λ1

, λ₁, λ₁

− λ1

2 χ⁰₁

V +V2

2 , λ2, λ2

−1 2χ⁰₂

V −V2

2λ₂ , λ2, λ2

=ψ

V +V₁

2 ,V −V₁ 2λ₁ , λ₁, λ₁

−ψ

V +V₂

2 ,V −V₂ 2λ₂ , λ₂, λ₂

= Ψ

(

^{V, V}1, λ²₁

)

⁻^Ψ

(

^{V, V}2, λ²₂

)

^.

Similarly, using the symmetries of Λ, we have

∂

∂V1

L(V, V₁, V₂, λ₁, λ₂) = Ψ

(

^V1, V, λ²₁

)

⁻^Ψ

(

^V1, V₂, λ²₁−λ²₂

)

and

∂

∂V2

L(V, V₁, V₂, λ₁, λ₂) =−Ψ

(

^V2, V, λ²₂

)

⁻^Ψ

(

^V2, V₁, λ²₁−λ²₂

)

^.

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 (3.1), arranged

as in Fig. 3(d).

The Lagrangian constructed this way is suitable for the continuum limit procedure, as the following proposition establishes.

Proposition 3.2. Every term of L is of at least first order in both parameters, L=O(λ1λ2).

Proof . Taking the limit λ1→0, the Lagrangian vanishes:

L(V, V, V₂,0, λ₂) =−λ₂χ

V +V₂

2 ,V −V₂ 2λ2

, λ₂, λ₂

+λ₂χ

V +V₂

2 ,V −V₂

−2λ2

,−λ₂, λ₂

= 0.

Similarly, for λ2→0 we have L(V, V1, V, λ1,0) = 0.

3.2 Q1_δ=0

A continuum limit for equation Q1,

λ²₁(V₂−V)(V₁₂−V₁)−λ²₂(V₁−V)(V₁₂−V₂) = 0,

with its pluri-Lagrangian structure, was given in [25]. The result is the hierarchy of Schwarzian KdV equations,

v2 = 0, v₃ v1

=−3v₁₁²

2v₁² +v₁₁₁ v1

, v4 = 0,

v₅

v₁ =−45v₁₁⁴

8v₁⁴ +25v²₁₁v₁₁₁

2v₁³ − 5v²₁₁₁

2v²₁ −5v₁₁v₁₁₁₁

v₁² +v₁₁₁₁₁ v₁ ,

· · ·

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wherevi is a shorthand notation for the derivativevti. Table1 shows how this continuum limit fits in the scheme of Section 3.1. We find the discrete Lagrangian

L=λ²_ilog

V −Vi

λi

−λ²_jlog

V −Vj

λj

−

(

λ²_i −λ²_j

)

log

Vi−Vj

λi−λj

.

Ψ

(

V, W, λ²

)

= λ² V −W ψ(v, v⁰, λ, µ) = µ

2v⁰

χ(v, v⁰, λ, µ) =µlog(v⁰) Λ(V, W, λ, µ) =λµlog

V −W 2λ

Table 1. Q1δ=0fact sheet. See Section 3.1for the meaning of these functions.

The first few coefficients of the continuous pluri-Lagrangian 2-form, after eliminating alien derivatives, are

L₁₂=−v₂ 4v₁, L₁₃= v²₁₁

4v²₁ − v₃ 4v1, L₁₄=−v₄

4v₁, L₁₅= 3v⁴₁₁

16v⁴₁ −v₁₁₁² 4v₁² − v₅

4v₁, L₂₃= v₁₁v₁₂

2v₁² +v²₁₁v₂

8v₁³ −v₁₁₁v₂ 4v₁² , L₂₄= 0,

L₂₅=−v₁₁₁v₁₁₂

2v₁² +3v₁₁³ v₁₂

4v₁⁴ −v₁₁v₁₁₁v₁₂

v₁³ +v₁₁₁₁v₁₂

2v²₁ +27v₁₁⁴ v₂ 32v⁵₁

−17v₁₁² v₁₁₁v₂

8v₁⁴ +7v₁₁₁² v₂

8v³₁ + 3v₁₁v₁₁₁₁v₂

4v₁³ −v₁₁₁₁₁v₂ 4v²₁ , L₃₄=−v₁₁v₁₄

2v²₁ − v²₁₁v₄

8v₁³ + v₁₁₁v₄ 4v₁² , L₃₅= 45v⁶₁₁

64v⁶₁ −57v₁₁⁴ v₁₁₁

32v⁵₁ +19v₁₁² v²₁₁₁

16v₁⁴ −7v₁₁₁³

8v₁³ +3v³₁₁v₁₁₁₁

8v⁴₁ +3v₁₁v₁₁₁v₁₁₁₁

4v³₁ −v²₁₁₁₁ 4v₁²

−3v₁₁² v₁₁₁₁₁

8v₁³ + v₁₁₁v₁₁₁₁₁

4v²₁ −v₁₁₁v₁₁₃

2v₁² +3v₁₁³ v₁₃

4v₁⁴ −v₁₁v₁₁₁v₁₃

v₁³ +v₁₁₁₁v₁₃

2v₁² −v₁₁v₁₅ 2v²₁ +27v₁₁⁴ v₃

32v⁵₁ −17v²₁₁v₁₁₁v₃

8v⁴₁ +7v²₁₁₁v₃

8v₁³ +3v₁₁v₁₁₁₁v₃

4v³₁ −v₁₁₁₁₁v₃

4v₁² −v₁₁² v₅

8v³₁ +v₁₁₁v₅ 4v²₁ , L₄₅=−v₁₁₁v₁₁₄

2v₁² +3v₁₁³ v₁₄

4v₁⁴ −v₁₁v₁₁₁v₁₄

v₁³ +v₁₁₁₁v₁₄

2v²₁ +27v₁₁⁴ v₄ 32v⁵₁

−17v₁₁² v111v4

8v₁⁴ +7v₁₁₁² v4

8v³₁ + 3v11v1111v4

4v₁³ −v11111v4

4v²₁ .

The even-numbered times correspond to trivial equations, v_2k = 0, restricting the dynamics to a space of half the dimension. We can also restrict the pluri-Lagrangian formulation to this space:

L=X

i<j

L_2i+1,2j+1dt2i+1∧dt2j+1

(12)

is a pluri-Lagrangian 2-form for the hierarchy of nontrivial Schwarzian KdV equations v₃

v1

=−3v₁₁²

2v₁² +v₁₁₁ v1

, v₅

v₁ =−45v₁₁⁴

8v₁⁴ +25v²₁₁v₁₁₁

2v₁³ − 5v²₁₁₁

2v²₁ −5v₁₁v₁₁₁₁

v₁² +v₁₁₁₁₁ v₁ ,

· · · 3.3 Q1_δ=1

Equation Q1_δ=1 reads

λ²₁(V2−V)(V12−V1)−λ²₂(V1−V)(V12−V2) +λ²₁λ²₂

(

λ²₁−λ²₂

)

= 0.

We apply the procedure of Section 3.1 to find a Lagrangian that is suitable for the continuum limit. Intermediate steps are listed in Table2.

Ψ

(

^{V, W, λ}²

)

^{= log}

V −W +λ² V −W −λ²

ψ(v, v⁰, λ, µ) = log

2v⁰+µ 2v⁰−µ

χ(v, v⁰, λ, µ) = (2v⁰+µ) log(2v⁰+µ)−(2v⁰−µ) log(2v⁰−µ)

Λ(V, W, λ, µ) = (V −W +λµ) log(V −W +λµ)−(V −W −λµ) log(V −W −λµ) Table 2. Q1δ=1fact sheet. See Section 3.1for the meaning of these functions.

In the continuum limit of the equation we find v₃ =v₁₁₁− 3

2

v₁₁² −¹₄ v₁ , v₅ =−45v⁴₁₁

8v³₁ +25v₁₁² v₁₁₁

2v²₁ −5v₁₁₁²

2v₁ −5v₁₁v₁₁₁₁

v₁ +v₁₁₁₁₁+25v₁₁²

16v₁³ −5v₁₁₁ 8v²₁ − 5

128v₁³,

· · ·

and v_2k = 0 for allk∈N. Some coefficients of the continuous pluri-Lagrangian 2-form are L₁₃= v²₁₁

2v²₁ − v₃ 2v1

+ 1 8v²₁, L₁₅= 3v⁴₁₁

8v⁴₁ −v²₁₁₁ 2v²₁ − v₅

2v1

− 5v₁₁² 16v₁⁴ − 1

128v⁴₁, L₃₅= 45v⁶₁₁

32v⁶₁ −57v₁₁⁴ v₁₁₁

16v⁵₁ +19v₁₁² v²₁₁₁

8v⁴₁ −7v₁₁₁³

4v₁³ +3v³₁₁v₁₁₁₁

4v⁴₁ +3v₁₁v₁₁₁v₁₁₁₁

2v³₁ −v²₁₁₁₁ 2v₁²

−3v₁₁² v₁₁₁₁₁

4v₁³ + v₁₁₁v₁₁₁₁₁

2v²₁ −v₁₁₁v₁₁₃

v₁² +3v₁₁³ v₁₃

2v₁⁴ −2v₁₁v₁₁₁v₁₃

v₁³ +v₁₁₁₁v₁₃ v₁²

−v₁₁v₁₅

v₁² +27v⁴₁₁v₃

16v₁⁵ −17v₁₁² v₁₁₁v₃

4v₁⁴ +7v²₁₁₁v₃

4v₁³ +3v₁₁v₁₁₁₁v₃

2v³₁ − v₁₁₁₁₁v₃

2v²₁ − v²₁₁v₅ 4v₁³ +v₁₁₁v₅

2v²₁ − 95v⁴₁₁

128v⁶₁ +41v₁₁² v₁₁₁

32v₁⁵ −27v₁₁₁²

32v₁⁴ −3v₁₁v₁₁₁₁

16v₁⁴ +3v₁₁₁₁₁

16v³₁ −5v₁₁v₁₃

8v₁⁴ −15v₁₁² v₃ 32v₁⁵ +5v₁₁₁v₃

16v⁴₁ + v₅

16v³₁ + 55v²₁₁

512v⁶₁ −25v₁₁₁

256v₁⁵ + 3v₃

256v₁⁵ − 5 2048v⁶₁.

(13)

3.4 Q2 For the equation

!

ψ(v, v⁰, λ, µ) = log

(2v+λµ)(2v⁰+µ) (2v−λµ)(2v⁰−µ)

χ(v, v⁰, λ, µ) = 1

λ(2v+λµ) log(2v+λµ) + (2v⁰+µ) log(2v⁰+µ)

−1

λ(2v−λµ) log(2v−λµ)−(2v⁰−µ) log(2v⁰−µ) Λ(V, W, λ, µ) = (V +W +λµ) log(V +W +λµ) + (V −W +λµ) log

V −W λ +µ

−(V +W −λµ) log(V +W −λµ)−(V −W −λµ) log

V −W λ −µ

Table 3. Q2 fact sheet. See Section3.1for the meaning of these functions.

The continuum limit hierarchy is v₃ =v₁₁₁− 3

2

v₁₁² −¹₄ v₁ −3

2 v₁³ v², v5 =−45v⁵₁

8v⁴ +15v₁³v₁₁

v³ +15v₁v₁₁²

4v² −45v⁴₁₁

8v³₁ −15v₁²v₁₁₁

2v² +25v₁₁² v₁₁₁

2v₁² −5v₁₁₁² 2v₁

−5v₁₁v₁₁₁₁ v1

+v₁₁₁₁₁+ 5v₁

16v² + 25v²₁₁

16v³₁ −5v₁₁₁ 8v₁² − 5

128v³₁,

· · ·

and v_2k = 0 for allk∈N. A few coefficients of the pluri-Lagrangian 2-form are L₁₃= 3v²₁

2v² + v₁₁² 2v₁² − v₃

2v1

+ 1 8v₁², L₁₅= 15v⁴₁

8v⁴ −15v₁₁²

4v² +3v⁴₁₁ 8v₁⁴ − v₁₁₁²

2v²₁ − v₅ 2v1

+ 5

16v² − 5v²₁₁ 16v⁴₁ − 1

128v₁⁴, L₃₅= 45v⁶₁

32v⁶ −9v₁⁴v₁₁

4v⁵ +63v₁²v²₁₁

32v⁴ −81v₁₁³

4v³ + 495v₁₁⁴

32v²v₁² +45v⁶₁₁

32v₁⁶ − 9v³₁v₁₁₁

16v⁴ +39v₁v₁₁v₁₁₁ 2v³

−165v₁₁² v₁₁₁

8v²v₁ −57v⁴₁₁v₁₁₁

16v₁⁵ +3v₁₁₁²

8v² +19v₁₁² v₁₁₁²

8v⁴₁ −7v₁₁₁³

4v₁³ −3v₁²v₁₁₁₁

v³ +27v₁₁v₁₁₁₁ 4v² +3v₁₁³ v₁₁₁₁

4v₁⁴ +3v₁₁v₁₁₁v₁₁₁₁

2v₁³ −v₁₁₁₁²

2v₁² −3v₁v₁₁₁₁₁

4v² −3v₁₁² v₁₁₁₁₁

4v₁³ +v₁₁₁v₁₁₁₁₁ 2v₁²

−v₁₁₁v₁₁₃

v²₁ −15v₁₁v₁₃

2v² +3v³₁₁v₁₃

2v₁⁴ −2v₁₁v₁₁₁v₁₃

v³₁ + v₁₁₁₁v₁₃

v₁² − v₁₁v₁₅

v₁² +75v₁³v₃ 16v⁴