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

Transcription

1 A variational perspective on continuum limits of ABS and lattice GD equations Mats Vermeeren Technische Universität Berlin arxiv:80855v [nlinsi] 5 Nov 08 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 Introduction This paper continues the work started in [3], 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, we review the continuum limit procedure from [3] 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 [] The code is available at

2 U 3 U 3 U 3 U 3 α 3 U α U U α U Figure : A quad equation is consistent around the cube if U 3 can be uniquely determined from U, U, U and U 3 If in addition U 3 is independent of U, then the equation satisfies the tetrahedron property The discrete pluri-lagrangian principle Most of the lattice equations we will consider are quad equations, difference equations of the form QU, U, U, U, α, α = 0, where subscripts of the field U : Z C denote lattice shifts, U Um, n, U Um +, n, U Um, n +, U Um +, n +, and α i C are parameters associated to the lattice directions Even though the equations live in Z, we require that we can consistently implement them on every square in a higher-dimensional lattice Z N This property of multidimensional consistency is an important attribute of integrability for lattice equations, see for example [3], [5], or [0] A necessary and sufficient condition for multidimensional consistency is that the equation is consistent around the cube: Definition Given lattice parameters α, α, and α 3 and field values U, U, U, and U 3, we can use the equations QU, U i, U j, U ij, λ i, λ j = 0, i < j 3 to determine U, U 3, and U 3 Then we can use each of the three equations QU i, U ij, U ik, U ijk, λ j, λ k = 0, i, j, k even permutation of,, 3 to determine U 3 If these three values agree for all initial conditions U, U, U, and U 3 and all parameters α, α, and α 3, then the equation is consistent around the cube Multidimensionally consistent equations on quadrilateral graphs were classified by Adler Bobenko and Suris in [3] The list of equations they found is widely known as the

3 Figure : Visualization of a discrete -surface in Z 3 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 [3] and was explored further in [4] Consider the lattice Z N with basis vectors e,, 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 e i + ε e j ε, ε {0, } Z N, where n = n,, 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 = Un, Un + e i, Un + e j, Un + e i + e j Occasionally we will also consider the corresponding filled-in squares in R N, { } i,j n = n + µ e i + µ e j µ, µ [0, ] R N, on which we consider the orientation defined by the -form dt i dt j The role of a Lagrange function is played by a discrete -form LU 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 LU ji n, λ j, λ i = L ij n, λ i, λ j Consider a discrete surface Γ = { α } in the lattice, ie a set of quads, such that the union of the corresponding filled-in squares α α is an oriented topological -manifold possibly with boundary The action over Γ is given by S Γ = i,j n Γ LU i,j n, λ i, λ j Definition 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 3

4 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 ηu, U i, λ i be a discrete -form Then every field U : Z N C is critical for the discrete pluri-lagrangian -form L = η, defined by LU, U i, U j, U ij, λ i, λ j = ηu, U i, λ i + ηu i, U ij, λ j ηu j, U ij, λ i ηu, 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 [3], which introduce the pluri-lagrangian or Lagrangian multiform idea, and to the reviews [5] and [0, Chapter ], 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 -form in R N, depending on a field u : R N C and any number of its derivatives Definition 4 The field u solves the continuous pluri-lagrangian problem for L if for any embedded surface Γ R n and any variation δu which vanishes near the boundary Γ, there holds δ L = 0 Γ 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 δ ij L ij δu I = 0 I t i, t j, δ ij L ij δu Itj δ ikl ik δu Itk = 0 I t i, δ ij L ij δu Iti t j + δ jkl jk δu Itj t k + δ kil ki δu Itk t i = 0 I, 3 for i, j, and k distinct, where δ ij δu I = α,β=0 α+β dα dt α i d β dt β j u It α i t β j 4

5 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, ie 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 has α j additional derivatives with respect to t i and β additional derivatives with respect to t j We call Equations 3 the multi-time Euler-Lagrange equations They were derived in [0] Continuum limit procedure In [3] 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,, n N into the continuous multi-time R N t,, t N as follows We associate a parameter λ i with each lattice direction and set t i = i+ λ i n i + + n λ i N N i 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, t i = i+ cλ i cλ i N n + + n N i i + τ i, 4 for constants c, τ,, 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 [6] We will call Equation 4 the Miwa correspondence We denote the shift of U in the i-th lattice direction by U i If Un = ut,, t N, It is given by U i = Un + e i = u t + cλ i, t cλ i,, t n N cλn 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 t -derivatives in the leading order, because only in the t -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, 5

6 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 [3] Consider N pairwise distinct lattice parameters λ,, λ N and denote by e,, 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 i v i = cλ i, cλ i,, N+ cλn i for i =,, N 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 ut, ut + v i, ut + v i + v j, We define L disc [u], λ, λ = L disc u, u + u + u +, u + u + u +, u + u + u + u + u + u +, λ, λ, where the differential operators correspond to the lattice directions under the Miwa correspondence, N k = j+ cλj k d j dt j j= 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 L Miwa [u], λ, λ = i,j=0 B i B j i! j! i j L disc[u], λ, λ, which is a formal power series in λ and λ Then by construction we have the formal equality L disc n = L Miwa η η, where η,, η N are the -forms dual tot the Miwa shifts v,, v N and 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 -form L = L Miwa η i η j i<j N This is the continuum limit of the discrete pluri-lagrangian -form, but it can be written in a more convenient form in terms of the coefficients of the power series L Miwa 6

7 Theorem 5 Let L disc be a discrete Lagrangian -form, such that every term in the corresponding power series L Miwa is of strictly positive degree in each λ i, ie such that L Miwa is of the form L Miwa [u], λ, λ = i+j c λi i i,j= λ j j L i,j[u] 5 Then the differential -form L = L i,j [u] dt i dt j, i<j N is a pluri-lagrangian structure for the continuum limit hierarchy, restricted to R N 3 Eliminating alien derivatives Suppose a pluri-lagrangian -form in R N produces multi-time Euler-Lagrange equations of evolutionary type, u tk = f k [u] for k {, 3,, N} If this is the case, then the differential consequences of the multi-time Euler-Lagrange equations can be written in a similar form, u I = f I [u] with I t k for some k {, 3,, N}, 6 where the multi-index I in f I is a label, not a partial derivative In this context it is natural to consider t 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, ie if I t k k {, i, j} If u I is not {i,, j}-native, ie if there is a k {, 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 -form with these coefficients by L A priori there is no reason to believe that the -form L will be equivalent to the original pluri-lagrangian -form L, but using the multi-time Euler-Lagrange equations one can derive the following result [3] 7

8 Theorem 7 If for all j the coefficient L j [u] does not contain any alien derivatives, then every critical field u for the pluri-lagrangian -form L is also critical for L In practice, given a Lagrangian -form, we can make it fulfill the conditions on the L j [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 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 we present this general strategy In Sections 3 37 we will discuss each individual equation of type Q 3 Three leg forms and Lagrangians All quad equations QV, V, V, V, α, α = 0 from the ABS list have a three leg form: QV, V, V, V, α, α = ΨV, V, α ΨV, V, α ΦV, V, α α 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: QV, V, V, V, λ, λ = ΨV, V, λ ΨV, V, λ ΨV, V, λ λ, where we introduced new parameters by λ 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, ie to consider QV, V, V, V,, λ, λ = ΨV, V, λ ΨV, V, λ ΨV, V,, λ λ, and to write the Ψ in terms of difference quotients We will introduce a function ψv, v, λ, µ = ψ v, λ, µ + ψ v, λ, µ of the continuous variables, from which we can recover ΨV, 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 = in the Miwa correspondence, which means that the derivative v t is approximated by difference quotients such as V V λ and V V λ We identify V + W ΨV, W, λµ = ψ, V W λ, λ, µ 8

9 All equations of the Q-list can be written in the form V + V QV, V, V, V,, λ, λ = ψ, V V, λ, λ λ V + V ψ, V V, λ, λ λ V + V, ψ, V V, λ λ, λ λ, λ + λ 7 To achieve the symmetries required of a quad equation, we require that ψ v, λ, µ = ψ v, λ, µ, ψ v, λ, µ = ψ v, λ, µ, ψ v, λ, µ = ψ v, λ, µ Furthermore, we require that ψv, v, 0, 0 = 0 We would expect the first nonzero terms in the series expansion at first order in λ, λ, but one can easily check that QV, V, V, V,, λ, λ = Oλ + λ 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, 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 3b: the difference quotient V, V can be expanded λ λ in a double power series, but its upward analogue V, V λ +λ cannot To find a Lagrangian for Equation 7, we follow [3, 4] and integrate the leg function ψ We take χ v, λ, µ = ψ v, λ, µ dv and χ v, λ, µ = ψ v, λ, µ dv λ Then and χ = χ + χ satisfies χ v, λ, µ = χ v, λ, µ, χ v, λ, µ = χ v, λ, µ = χ v, λ, µ, Now λ v χv, v, λ, µ + v χv, v, λ, µ = ψv, v, λ, µ V + W ΛV, W, λ, µ = λ χ, V W λ, λ, µ 9

10 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 gives the terms of the Lagrangian in triangle form: LV, V, V, λ, λ = ΛV, V, λ, λ ΛV, V, λ, λ ΛV, V, λ λ, λ + λ V + V = λ χ, V V V + V, λ, λ λ χ, V V, λ, λ λ λ V + V V V λ λ χ, λ λ, λ λ, λ + λ 8 Note the symmetries of Λ: ΛV, W, λ, µ = ΛW, V, λ, µ = ΛV, 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 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 V LV, V V +, V, λ, λ = λ V χ V, V V, λ, λ λ V + λ V χ V, V V, λ, λ λ = λ V + V χ, λ, λ + V V χ, λ, λ λ λ V + V χ, λ, λ V V χ, λ, λ λ V + V = ψ, V V V + V, λ, λ ψ, V V, λ, λ λ λ = ΨV, V, λ ΨV, V, λ 0

11 Similarly, we have V LV, V, V, λ, λ = ΨV, V, λ ΨV, V, λ λ and LV, V, V, λ, λ = ΨV, V, λ V ΨV, V, λ λ 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 3d 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 = Oλ λ Proof Taking the limit λ 0, the Lagrangian vanishes: V + V LV, V, V, 0, λ = λ χ, V V V + V, λ, λ + λ χ, V V, λ, λ λ λ = 0 Similarly, for λ 0 we have LV, V, V, λ, 0 = 0 3 Q δ=0 A continuum limit for equation Q, λ V V V V λ V V V V = 0, with its pluri-lagrangian structure, was given in [3] The results is the hierarchy of Schwarzian KdV equations, v = 0, v 3 = 3v v v + v, v v 4 = 0, v 5 = 45v4 v 8v 4 + 5v v v 3 5v v 5v v v + v, v 9 where v i is a shorthand notation for the derivative v ti Table shows how this continuum limit fits in the scheme of Section 3 We find the discrete Lagrangian V L = λ Vi V i log λ Vj j log λ i λ Vi V j λ i λ j log j λ i λ j

12 ΨV, W, λ = λ V W ψv, v, λ, µ = µ v χv, v, λ, µ = µ logv V W ΛV, W, λ, µ = λµ log λ Table : Q δ=0 fact sheet See Section 3 for the meaning of these functions The first few coefficients of the continuous pluri-lagrangian -form, after eliminating alien derivatives, are L = v 4v L 3 = v 4v L 4 = v 4 4v L 5 = 3v4 6v 4 L 3 = v v L 4 = 0 v v 3 4v v 4v L 5 = v v v + v v 8v 3 7v v v 8v 4 L 34 = v v 4 v L 35 = 45v6 64v 6 v 5 4v v v 4v + 3v3 v 4v 4 v v 4 8v 3 57v4 v 3v 5 3v v 8v 3 + 7v4 v 3 3v 5 L 45 = v v 4 v 7v v v 4 8v 4 + 7v v 8v 3 v v v v 3 + v v 4 4v + 9v v 6v 4 + v v 4v 7v v v 3 8v 4 + 3v3 v 4 4v 4 + 7v v 4 8v 3 + 3v v v 4v 3 + v v v 7v3 8v 3 v v 3 v + 7v v 3 8v 3 v v v 4 v 3 + 3v v v 4 4v 3 v v 4v + 7v4 v 3v 5 + 3v3 v 8v 4 + 3v3 v 3 4v 4 + 3v v v 3 4v 3 + v v 4 v v v 4 4v + 3v v v 4v 3 v v v 3 v 3 v v 3 4v + 7v4 v 4 3v 5 + v v 3 v v v 5 8v 3 v 4v v v 5 v + v v 5 4v The even-numbered times correspond to trivial equations, v ti = 0, restricting the dynamics to a space of half the dimension We can also restrict the pluri-lagrangian

13 formulation to this space: L = i<j L i+,j+ dt i+ dt j+ is a pluri-lagrangian -form for the hierarchy of nontrivial Schwarzian KdV equations, v 3 = 3v v v + v, v v 5 = 45v4 v 8v 4 + 5v v v 3 5v v 5v v v + v, v 33 Q δ= Equation Q δ= reads λ V V V V λ V V V V + λ λ λ λ = 0 We apply the procedure of Section 3 to find a Lagrangian that is suitable for the continuum limit Intermediate steps are listed in Table V W + λ ΨV, W, λ = log V W λ v ψv, v + µ, λ, µ = log v µ χv, v, λ, µ = v + µ logv + µ v µ logv µ ΛV, W, λ, µ = V W + λµ logv W + λµ V W λµ logv W λµ Table : Q δ= fact sheet See Section 3 for the meaning of these functions In the continuum limit of the equation we find v 3 = v 3 v 4 v, v 5 = 45v4 8v 3 + 5v v v 5v 5v v + v + 5v v v 6v 3 5v 8v 5 8v 3, and v k = 0 for all k N Some coefficients of the continuous pluri-lagrangian -form are L 3 = v v v 3 v + 8v 3

14 L 5 = 3v4 8v 4 v v v 5 5v v 6v 4 L 35 = 45v6 3v 6 34 Q 57v4 v 6v 5 + v v v + 7v v 3 4v 3 + 3v 6v 3 For the equation v v 3 v + 3v v v 3 v 3 5v v 3 8v 4 + 9v v 8v 4 + 3v3 v 3 v 4 v v 3 v 5v v 3 3v 5 8v 4 7v3 4v 3 + 3v3 v 4v 4 v v v 3 v 3 v v 5 4v 3 + 5v v 3 6v 4 + v v 5 v + 3v v v v 3 + v v 3 v v v 5 v v v + 7v4 v 3 6v 5 3v v 4v 3 7v v v 3 4v 4 95v4 + 4v 8v 6 v 7v 3v 5 3v v 3v 4 6v 4 + v v 6v 3 5v 5v 6 + 3v 56v v 5 048v 6 λ V V V V λ V V V V +λ λ λ λ V + V + V + V λ 4 + λ λ λ 4 = 0 the general strategy of Section 3 works with the choices listed in Table 3 V + W + λ ΨV, W, λ V W + λ = log V + W λ V W λ v + λµv ψv, v + µ, λ, µ = log v λµv µ χv, v, λ, µ = λ v + λµ logv + λµ + v + µ logv + µ λ v λµ logv λµ v µ logv µ V W ΛV, W, λ, µ = V + W + λµ logv + W + λµ + V W + λµ log + µ λ V W V + W λµ logv + W λµ V W λµ log µ λ Table 3: Q fact sheet See Section 3 for the meaning of these functions The continuum limit hierarchy is v 3 v 3 = v 3 v 4 3 v v, v 5 = 45v5 8v 4 + 5v3 v v 3 + 5v v 4v 45v4 8v 3 5v v v + 5v v v 5v v 5v v + v + 5v v 6v + 5v 6v 3 5v 8v 5 8v 3, 4

15 and v k = 0 for all k N A few coefficients of the pluri-lagrangian -form are L 3 = 3v v + v v v 3 v + 8v L 5 = 5v4 8v 4 5v 4v + 3v4 8v 4 v v v v 6v 5v 6v 4 8v 4 L 35 = 45v6 9v4 3v 6 v + 63v 4v 5 v 8v3 3v v4 4v v6 3v v 9v3 3v 6 v + 39v v v 65v 6v 4 v 3 v 8v v 57v4 v 6v 5 v v + v v 3 v + 7v v 3 4v 3 + 3v + 9v 8v v 7v3 8v 4 3v 4v 3 v + 7v v + 3v3 v 3 4v v + 3v v v 4v 4 v 3 3v v 3v 4v v + v v v v 3 5v v 3 + 3v3 4v 3 v v v v 3 v v v 3 v 4 v 3 v v 5 v + 3v v v 3 v v3 v 3 5v v v 3 + 5v 6v 4 v 3 v 3 8v v + 7v4 v 3 + 5v v 3 7v 6v 5 4v v v 3 4v 4 v v 3 v 9v v 5 v 4v v 5 + v v 5 35v 4v 3 v + 65v 8v 4 6v 3 55v 95v4 64v v 35v 8v 6 3v v + 4v v 7v 3v 5 3v v + 3v 3v 4 6v 4 5v v 3 6v 3 8v 4 + 5v 3 3v v 5v v 3 + 5v v 3 + v 3v 5 6v v 6v 3 5v v 5v 5v 6 + 3v 56v v 5 048v 6 35 Q3 δ=0 We consider equation Q3 in the form sinλ e iv e iv + e iv e iv sinλ e iv e iv + e iv e iv The strategy of Section 3 works as outlined in Table 4 ΨV, W, λ e iλ e iv e iw = log e iv e iλ e iw V W + λ = log sin ψv, v, λ, µ = log sin λv + λµ χv, v, λ, µ = i λ sinλ λ e iv e iv + e iv e iv = 0 log sin V W λ log sin λv λµ λ µv + Li e iλv +λµ Li e iλv λµ ΛV, W, λ, µ = λµv W Li e iv W +λµ iv W + Li e λµ Table 4: Q3 δ=0 fact sheet See Section 3 for the meaning of these functions 5

16 The continuum limit hierarchy is v 3 = v 3 v 4 v + v3, v 5 = 3 8 v5 5 4 v v + 5 v v 5 48 v 45v4 8v 3 + v + 5v 6v 3 5v 8v 5 8v 3, + 5v v v 5v v 5v v v and v k = 0 for all k N Some coefficients of the continuous pluri-lagrangian -form are L 3 = 4 v + v 4v v 3 4v + 6v L 5 = 6 v v + 3v4 6v 4 v 4v v 5 4v 5v 3v 4 56v L 35 = 64 v v v v3 v 5 3 v3 v v 65v4 + 55v 64v v 6v 6 v 9 8 v v + 8 v v v v 3 5v v 3 6v 5 8 v v v v v + 45v6 8v + 35v 64v 6 9v 57v4 v 3v 5 + 9v v 6v 4 + 3v3 v 3 4v 4 + 3v v v 3 4v 3 + 3v 3v 3 7v3 8v 3 v v v 3 v 3 v v 3 4v 5v v 3 6v 4 + 3v3 v 8v 4 + v v 3 v v v 5 8v 3 5v v 3 64v 5 + 3v v v 4v 3 v v 5 v + v v 5 4v + 5v v 3 3v 4 v 4v 3v v 8v 3 + v v 4v v v 3 v 5v 3 9v + 7v4 v 3 7v 3v 5 v v 3 + 7v 8v 4 v 3 8v v4 04v + 4v 56v 6 v 7v 64v 5 3v v 64v 4 3v 4 + v v 3v 3 5v 04v 6 + 3v 5v v v 6 Note on dilogarithms In Table 4 we encounter the dilogarithm function [5], given by Li z = n= z n n for z < and by analytic continuation for other z C \ [, Its derivative is d dz Li log z z = z This can be used to integrate log sin, which occurs in the leg function Ψ, logsin z dz = i z + Li e iz z log i + c 6

17 In the continuum limit procure we need a series expansion of Li e x For small x this seems problematic, since Li z has a branch point at z = The way to handle this is to carry along this branch point in the formal series expansion Li e x = Li x log x + x x 4 x3 7 +, 0 where the first term famously equals Li = π 6 36 Q3 δ= For equation Q3 with nonzero parameter we use a slightly different form compared to Q3 δ=, sinλ cosv cosv + cosv cosv sinλ cosv cosv + cosv cosv sinλ λ cosv cosv + cosv cosv sinλ sinλ = 0 The general strategy of Section 3 applies to this transformed equation with the choices in Table 5 V W + λ ΨV, W, λ = log sin log sin ψv, v, λ, µ = log sin λv + λµ χv, v, λ, µ = i λ sin V + W + λ V W λ V + W λ sin sin v + λµ log sin λv λµ sin v λµ λ µv + Li e iλv +λµ Li e iλv λµ i λµv + Li e iv+λµ Li e iv λµ λ ΛV, W, λ, µ = λµv W Li e iv W +λµ iv W + Li e λµ + λµv + W Li e iv +W +λµ iv +W + Li e λµ Table 5: Q3 δ= fact sheet See Section 3 for the meaning of these functions 7

18 The continuum limit hierarchy is v 3 = v 3 v 4 + v v3 3 sinv, v 3 v 5 = 3 8 v5 5 4 v v + 5 v v + 5v5 4 sin v + 5v3 v cos v sin v v 45v4 8v 3 + 5v v v 5v 5v v + v 45v5 v v 8 sin v 4 + 5v v 4 sin v 5v v sin v + 5v 6v 3 5v 8v + 5v 6 sin v 5 8v 3, and v k = 0 for all k N A few coefficients of the continuous pluri-lagrangian -form are L 3 = 4 v + v 4v v 3 4v + 3v 4 sin v + 6v L 5 = 6 v v 5 vv 3 5 v v 3 5v4 8 sin v + 3v4 6v 4 v 5 4v + 5v4 6 sin v 4 5v 8 sin v 5v 3v 4 + v 4v 5 3 sin v 56v L 35 = 3 9 v v v v3 v v3 v v v4 64 sinv v cosv + 8 sinv v 05v4 64v + 35v v 6v v 9 8 v v + 8 v v v v 3 + 5v v 3 6v 35 4 v v 3 5 vv v v 5 + 5v6 + 39v 64 sinv 4 v 7v3 3 sinv v 5v3 6 sinv v 3 9v4 6 sinv v cosv 8v3 8 sinv 5 cosv 8 sinv v v v cosv 3v 4 sinv 3 v cosv 5v v v 3 cosv + 5v sinv 3 4 sinv v6 8v + 95v 64v 6 9v 57v4 v 3v 5 + 9v v 6v 4 + 3v3 v 3 4v 4 + 3v v v 3 4v 3 7v3 8v 3 v v v 3 v 3 v v 3 4v + 3v3 v 8v 4 + v v 3 v v v 5 8v 3 + 3v v v 4v 3 v v 5 v + v v 5 4v v 4v 3v v 8v 3 + v v 4v v v 3 v 65v 3 9v + 7v4 v 3 7v 3v 5 v v 3 + 7v 8v 4 v 3 8v v6 64 sinv v v 64 sinv 4 9v3 v 3 sinv v3 v 3 3 sinv 4 5v + 495v4 8 sinv 65v 64v v + 3v sinv 6v sinv + 7v v 3v v 5v v 3 6 sinv 8 sinv 8 sinv 4 sinv + 5v v 3 + 5v v 3 9v v v cosv v4 6v sinv 8 sinv 8 sinv 3 sinv 3 04v + 4v 56v 6 v 7v 64v 5 64v 4 3v v 3v 4 + 3v 3v 3 5v v 3 6v 4 5v v 3 64v 5 + 5v v 3 3v 4 + v 5 35v 3v 3 55v 56 sinv 4 8v sinv 35v 5v v 64v sinv 64v sinv 5v 04v 6 + 3v 5v v 5 04v sinv 4096v 6 8

19 37 Q4 Consider equation Q4 in its elliptic form, A V b V b a bc b V b V b a bc b + B V a V a b ac a V a V a b ac a = ABCa b, where a, A = λ, λ, b, B = λ, λ, c, C = λ λ, λ λ, and is the Weierstrass elliptic function The approach of Section 3 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 and g 3 of the - function They satisfy ζ = σ σ, g =, ζ =, g 3 = Additionally, for the series expansion of ψ we use g m σz = a m,n g3 n z 4m+6n+ 4m + 6n +!, m,n=0 where the first coefficients are a 0,0 =, a,0 =, and a 0, = 3 These and many more identities involving the Weierstrass elliptic functions can be found in [, Chapter 8] The continuum limit hierarchy is v 3 = 6v 3 v 3v v + v + 3 8v v 5 = 90v 5 v + 44v 5 v 4v 5 v + 60v3 v v v v 60v3 v v v v + 60vv 3 v + 5v v v 30vv v v v 45v4 8v 3 5v 5v v + v + 5v v v 6v 3 5v 8v 5 8v 3 + 5v v v and v k = 0 for all k N To simplify these equations we used the doubling formula for the Weierstrass function, v v = v + v 9

20 σv + W + λ ΨV, W, λ σv W + λ = log σv + W λ σv W λ σv + λµσλv ψv, v + λµ, λ, µ = log σv λµσλv λµ = ζvλµ 3 vλ 3 µ v vλ 5 µ logv + µ g λv + λµ 4 40 g 3λv + λµ logv µ g λv λµ 4 g 3λv λµ χv, v, λ, µ = logσvλµ λ 3 vλ3 µ 3 0 v λ 5 µ v + µlogv + µ g λv + λµ λ v µlogv µ g λv λµ λ ΛV, W, λ, µ = logσv + W λµ 3 V + W λ3 µ 3 0 V + W λ 5 µ 5 + V W + V W + µλ log + µ g V W + λµ 5 + λ 00 V W V W µλ log µ + g V W λµ 5 + λ 00 Table 6: Q4 fact sheet See Section 3 for the meaning of these functions As the first nontrivial equation of the limit hierarchy we recognize the Krichever- Novikov equation [], [8] 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 [] The continuous pluri-lagrangian -form has the coefficients L 3 = 3v v + v 4v v 3 4v + 6v L 5 = 5v 4 v 4v 4 v + 4v 4 v 5 v v + 3v4 6v 4 v 4v v 5 4v 5v 3v 4 56v v 0

21 L 35 = 45v 6 v3 + 6v 6 v v 88v 6 v3 36v 6 v v + 7v 6 v v 36v 6 v 8v4 v v v v + 8v4 v v v v v 8v 4v v v v v v 9 v3 v v + 75 v3 v 3 v 54v v v + 36v 3v v 60v 3 v 3 v + 9v v v 6v 3 v v + 0v 3 v 3 v 35 6 v v + 5 v v 8v3 v v v 5v v v 3 v v v + 8v3 v v v + 39v v v v v v 39v v v v v v 6v v v v v + 6v v v v v + 5v v v 3 v v v 8 v3 v+39v v v v 6v v v 5v v v 3 v + 495v4 v 6v 65v v v 4v 5v v 3 v + 5v v 3 v 4v + 65v v v 8 v + 45v6 64v 6 + v v 4v + 7v v 3 8v 3 3v v 3v 4 57v4 v 3v 5 v v 3 v + 3v v v 3 4v 3 + 3v 3v 3 65v v v 8 v + 9v v 6v 4 + 3v3 v 3 4v 4 v v 3 4v 5v v 3 6v v v + 7 v v v 3 v v v + 5 v v 3 v 9 v v 5 v 7 4 v v v v 55v v 3v 7v3 8v 3 v v v 3 v 3 v v 5 8v 3 5v v 3 64v 5 4 ABS equations of type H + 3v3 v 8v 4 + v v 5 4v + 5v v 3 3v 4 + v v 3 v + 3v v v 4v v 56v v v 5 v 35v v 6v v 4v + 7v4 v 3 3v 5 + 5v 3 v 6v 3v v 8v 3 7v v v 3 8v 4 95v4 + 4v 56v 6 v 7v 64v 5 64v 4 + v v 3v 3 5v 04v 6 + 3v 5v v v 6 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 H was discussed in [3] 4 H No continuum limit is known for equation H, which reads U U U U + λ λ U + U + U + U + λ 4 λ 4 = 0 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 = n+m Un, m The resulting equation V V V V ± λ λ V V V + V + λ 4 λ 4 = 0 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

22 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, 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 H that treat both lattice directions equally [9]: and Un, m = Un, m = λ n + λ m + c n λ + m λ + c 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 4 H3 δ=0 Equation H3 is usually given as λ UU + U U λ UU + U U = 0, but it is useful to perform a nonautonomous transformation Un, m i n+m Un, m, which gives us λ UU U U λ UU U U = 0, 3 which is known as the lattice modified KdV equation [7] Equation 3 is the hierarchy The continuum limit of u 3 = u 3 u u u, u 5 = 0 u3 u u u u + u u u 5 u u + u u + u, u and u k = 0 for all k N However, we run in to trouble on the Lagrangian level The Lagrangian given in [3] for H3, adapted to the case δ = 0, is L = log UU λ log UU λ + Li λ U λ U λ U Li λ U + logλ logλ logu + logλ logu logu,

23 where Li is the dilogarithm function Unfortunately, the occurrence of expressions like logλ 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 V Then Equation 3 turns in to λ e i V +V e i V +V λ e i V +V e i V +V = 0 Multiplying with exp i 4 V + V + V + V turns this into λ sin 4 V + V V V λ sin 4 V V + V V = 0, 4 which is the form in which H3 δ=0 arises from Bcklund transformations for the Sine Gordon equation [0, p60] Additionally, it is only a simple transformation removed from the discrete Sine-Gordon equation of Hirota [] Using standard trigonometric identities we can rewrite the equation as V V V V λ λ tan λ + λ tan = 0, 4 4 which we can put in a three-leg form: λ + λ V V arctan tan λ λ 4 = V V 4 V V 4 From the three-leg form we can derive a Lagrangian as in [4] It takes the implicit form where L = 8 V V 8 V V I λ,λ V V, I λ,λ x = x 0 λ + λ y arctan tan dy λ λ 4 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 4 we find the potential modified KdV hierarchy, v 3 = v + v3, v 5 = 3 8 v5 + 5 v v + 5 v v + v, 3

24 and v k = 0 for all k N Some coefficients of the pluri-lagrangian -form are L 3 = 6 v4 4 v + 4 vv 3 L 5 = 3 v6 5 8 v v + 4 v + 4 vv 5 L 35 = 3 56 v v4 v 7 3 v5 v v5 v v4 7 8 v v v 3 8 v v v v v 8 v3 v 5 4 v v v v v v v v v 3 8 v3 v v 4 v v + v v 3 v v 3 + v v v v 3 4 v v 5 43 H3 δ= Due to difficulties analogous to those of equation H, no continuum limit is known for equation H3 δ=, λ UU + U U λ UU + U U + λ λ = 0 5 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 3 in a way that breaks the symmetry: Un, m Un, m m, λ λ, λ λ Setting again U = exp i V, we find the equation sin 4 V + V + V + V sin λ λ 4 V V V + V = 0 5 In order to recover some kind of multidimensional consistency for this equation, we consider an even-dimensional lattice Z N with coordinates n,, n N, n,, n N, where the coordinates n,, n N correspond to lattice directions where the above transformation has taken place Then on any quad spanned by directions n k and n l we impose Equation 5, sin 4 V + V k + V l + V kl sin λ k λ l 4 V V k V l + V kl = 0 6 The original equation H3, in the form of Equation 4, is imposed on quads spanned by directions n k and n l, and also on quads spanned by directions n k and n l, because transforming both directions leaves the equation invariant 4

25 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,, t N, t i = i+ cλ i cλ i N n + + n N + τ i, i i t i = i+ n cλ i i + + n N cλ i N i + τ i Then in the leading order of the continuum limit, Equation 6 produces the Sine- Gordon equation, sinv v = 0 In addition to this equation, the full series expansion in the parameters λ k and λ l yields two copies of the modified KdV hierarchy, v = 0, v 3 = v + v3, v 4 = 0, v 5 = 3 8 v5 + 5 v v + 5 v v + v, v = 0, v 3 = v + v3, v 4 = 0, v 5 = 3 8 v5 + 5 v v + 5 v v + v, 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 l, respectively n k and n l The discrete Lagrangian takes slightly different forms on the three types of quads: where and L kl = 8 V k V 8 V l V I λk,λ l V k V l, L kl = 8 V k V 8 V l + V J λk,λ l V k + V l, L kl = 8 V k + V 8 V l + V + I λk,λ l V k V l, I λk,λ l x = x J λk,λ l x = I λk, λ l x = 0 λk + λ l y arctan tan dy λ k λ l 4 x 0 λk λ arctan l + y λ k λ l tan dy 4 To get the required leading-order cancellation, we add to L the exterior derivative of the 5

26 discrete -form η given by ηv, V l = 8 V + V l and ηv, V k = 0 This yields L kl = 8 V k V 8 V l V I λk,λ l V k V l, L kl = 8 V k V 8 V l + V J λk,λ l V k + V l Vk 8 V + V kl V, l L kl = 8 V k + V 8 V l + V + I λk,λ l V k V l + V 4 l V k = 8 V k V + 8 V l V + I λk,λ l V k V l The resulting pluri-lagrangian -form is of the form L = L ij dt i dt j, i,j {,,,,} where L ij = L ji If i, j {,, }, then the coefficients are exactly those found in Section 4 If i, j {,, } 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, L = L ij dt i dt j Some coefficients of mixed type are i,j {,,3,3,} L = 4 v v cos v L 3 = 4 v cos v v v + 4 v 3v + v sin v L 5 = 3 6 v4 cos v 5 4 v v v v v sin v + 4 v cos v v v cos v + v v v v + 4 v 5v + v sin v L 3 = 4 v cos v v 3 v + v v 4 v v 3 + v sin v L 33 = 8 v v cos v + 4 v v sin v + 4 v v sin v + v v cos v v v 3 4 v 33 v v v + v 3 v cos v L 35 = 3 3 v4 v cos v v v v 3 sin v + 6 v4 v sin v + 8 v v cos v 4 v v v cos v v v v cos v 5 4 v v v 3 4 v3 v + 4 v v sin v 4 v v sin v + v v v sin v 4 v cos v + v v cos v + v v 3 v v 3 4 v 53 v v v + v 5 v + v sin v 6

27 L 5 = 7 44 v4 5 cos v + 36 v v3 + 5 v v v v v sin v + 4 v cos v 9 v v 5 cos v 8 v v 3 cos v v 5 v 8 v v v 3 v 9 v v 5 8 v 3 v 4 v v 5 8 v sin v v 3 L 53 = 7 88 v v4 cos v v v v v cos v 9 v v v v 3 v v v sin v 4 v v v sin v sin v v v v 5 cos v 36 v v v 3 cos v 9 v v 3 v 3 v v 4 v v sin v + 9 v v v sin v sin v v v 3 sin v v v v 3 sin v 4 v cos v 8 v v cos v v v 3 cos v v v 5 4 v 35 v 8 v 3 v v 33 v 9 v 3 v 5 8 v 33 v 9 v v 5 8 v v 3 + v sin v A few examples of multi-time Euler-Lagrange equations are 0 = δ L δv = v + sinv, 0 = δ 3 L 3 δv δ 33 L 33 δv 3 = 4 v = δ 3 L 3 δv δ 33 L 33 0 = δ 5 L 5 δv δ 5L 5 δv δv 3 = 4 v = 4 v 5 v 3 + v v 3, v 3 + v v 3, cos v 3 6 v v v v v + v 4 v 5 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 6 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 Proposition 0 If the difference equation Q = 0 satisfies QU, U, U, U, λ, λ = ±QU, U, U, U, λ, λ then the continuum limit hierarchy is invariant under simultaneous reversal of all evennumbered times, t k t k Proof Due to the symmetry assumptions on Q, we have QU, U, U, U, λ, λ = ±QU, U, U, U, λ, λ 7

28 Let ut, t, be a solution to the continuum limit equations Then Q ut, t,,u t + cλ, t c λ,, u t + cλ, t c λ,, u t + cλ + cλ, t c λ c λ,, λ, λ = 0 7 Now consider the same equation at times shifted once in both lattice directions, τ i = t i + i+ λ i c i + λi i Due to the symmetry of Q we have that Q uτ, τ,,u τ cλ, τ + c λ,, u τ cλ, τ + c λ,, u τ cλ cλ, τ + c λ + c λ,, λ, λ = 0 Introducing the parameters µ = λ and µ = λ we find Q uτ, τ,,u τ + cµ, τ + c µ,, u τ + cµ, τ + c µ,, u τ + cµ + cµ, τ + c µ + c µ,, µ, µ = 0 8 Comparing Equations 7 and 8, we immediately see that their series expansions in λ, λ respectively µ, µ only differ by a minus sign for each derivative with respect to an even-numbered time t k In other words, if we have a solution u to the continuum limit hierarchy, then reversing all even times gives a new solution Corollary If the continuum limit hierarchy of the difference equation Q = 0, as in Proposition 0, consists of evolutionary equations u k = f k u, u, u,, 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] 7 Gelfand-Dickey hierarchies A discrete counterpart of the Gelfand-Dickey hierarchy was introduced in [9], its pluri- Lagrangian structure in [4] The N-th member of the discrete hierarchy is a system of quad equations with N 3 components, which we denote by V [],, V [N ], W [],, W [N ], and U = V [0] = W [0] The equations are V [j+] V [j+] = + U U V [j] λ λ V [j] + V [j] λ λ, 9 W [j+] W [j+] = + U U W [j] W [j] + W [j] λ λ λ λ, 0 8

29 for j = 0,, N 3, and V [N ] W [N ] = λ N λ N λ γ N + λ + U U N 3 i=0 N 3 j=0 N 3 i j=0 γ N 3 i j V [j] W [i] γ N j V [j] W [j], where γ j = j + λ j λ j λ + + λ λ j + λ j It is important to note that the variables V [N ] and W [N ] can be eliminated The reduced system consists of equations 9 0 for j = 0,, N 4 and the 9-point equation + U U λ λ + + U U λ λ = λ N λ N λ + λ + U U λ N λ N V [N 3] V [N 3] + V [N 3] λ λ W [N 3] + W [N 3] W [N 3] λ λ N 3 i=0 λ λ + U U N 3 j=0 N 3 i j=0 γ N 3 i j V [j] W [i] V [j] W [i] γ N j V [j] W [j] V [j] + W [j], obtained by evaluating V [N ] W [N ] V [N ] +W [N ] once with Equations 9 0 and once with Equation The Lagrangian given in [4] for the N-th lattice Gelfand-Dickey equation is L = N+ log λ N λ U N λ λ + U γ N U U N + j=0 N 3 j=0 γ N j U U V [j] N N 3 j i=0 j= N j i=0 γ N i j W [i] V [j] V [j] γ N 3 i j W [i] λ λ U + U V [j] λ V [j] + λ V [j] Note that this Lagrangian depends on the field V [N ] but not on W [N ] A more symmetric equivalent Lagrangian is obtained by adding the exact discrete differential form UV [N ] U V [N ] + U V [N ] + UV [N ], 9

30 which does not change the Euler-Lagrange equations see Proposition 3 The resulting Lagrangian does not depend on V [N ] either, L = N+ log λ N λ U N λ λ + U γ N U U N 3 + j=0 N 3 j=0 γ N j U U V [j] N 3 N 3 j i=0 j= N j i=0 γ N i j W [i] V [j] V [j] γ N 3 i j W [i] λ λ U + U V [j] λ V [j] + λ V [j] Now that both variables V [N ] and W [N ] are absent from the Lagrangian, it becomes clear that the variational formulation does not produce the quad version of the Equation 9, but rather the 9-point version 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 3 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 7-point cube for the 9-point formulation For the Boussinesq equation the former was explicitly done in [] and the latter was numerically verified in [4, Section 57] The multidimensional consistency also follows from the construction of the GD hierarchy using the direct linearization method [9] In this construction, lattice shifts are identified with Bcklund transformations, hence the permutability property implies multidimensional consistency 7 Continuum limit of the lattice Boussinesq equation GD3 We consider the lattice Boussinesq equation in its 9-point scalar form, ie Equation for N = 3, λ 3 λ 3 λ λ + U U + + U U λ λ + λ λ λ 3 λ 3 λ U U + U U λ + U U U + U + U + + U U U λ λ λ + λ U + U = 0 For this equation we use the Miwa correspondence 4 with c = 3 perform a double series expansion of the lattice equation, F ij λ i λ j = 0 i,j=0 3 As always, we 30

31 The first column of coefficients of this expansion is F 00 = 0, F 0 = 8v v 9 v 3 v, F 0 = 8v + 8v v 8 4 v 7 4 v 3v 3, F 30 = 08v v + 648v v + 43v v + 54v v + 54v v 8v v 9v v 54v 7 v 9v v 3 9 v 3v v 33, F 40 = 80v v 405v v 35v v 35v v v + 5v v v v v v v + 5 v v + 45v v v v 35 v 35 4 v v + 45v v v v 35 v v 5v 3 v 35 4 v v 45 4 v v 5v v 3 5v v v 6 8 v 45 4 v 3 5v v 45 4 v 4 5v v 3 3v 5 5 v 34, From F 0 = 0 we get the equation v = v v 3v Using this equation we get v 3 = 0 from F 0 = 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 to find the hierarchy v = v v 3v, v 3 = 0, v 4 = 6v v + 3v, v 5 = 5v v + 45v v 5 4 v 9v, 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 LU, U, U, U, λ, λ = log λ 3 λ 3 + λ + λ λ + λ λ λ + U U U + U λ λ U U + + λ λ UU λ UU + λ UU U U U 3

32 In order to get the necessary leading order cancellation, we add the following terms to the Lagrangian, UU + U U U U UU + λ λ λ λ 3 U 3 U 3 + U U + U U + U U 4 λ λ λ + U U λ These terms do not contribute to the Euler-Lagrange equations because they are the discrete exterior derivative of the -form ηu, U i, λ i = λ i UU i + 3 U 3 + 4λ i U i U Some coefficients of the Lagrangian -form are L = v 3 + v 3 4 v L 3 = 3 v v 3 L 4 = 4v 4 v v + 6vv v 4 3 v v 3 v v 4 L 5 = 0v 3 v + 0v v 5v v 5 4 v v v 3 v v 5 L 3 = 6v v 3 v v 3 + v v 3 L 4 = 6v 3 v 8v v v 4v v + 8v v v + v 3 + 6vv v 6v v v v v v 4 v v + v v 4 L 5 = 6v v v v3 v 5v v v v 8 3 v v 64 3 v v v + 0v v + 0v v 0v v v + 5v v 0v v v 6v v v v v v 8 3 v v v v v v + v v 5 L 34 = 6v 3 v 3 4v v v 3 + 6vv v 3 + v v 3 + 4v v v v v v v 3 v v v v 3 L 35 = 30v v v 3 + 0v v 3 0v v 3 v 0v v v 3 + 0v v v 3 + 0v v v 3 + 0v v v v v v v v v v v v v 3 L 45 = 60 3 v6 60v 3 v 60v 4 v 40v 3 v + 76v v v v + 44v v + 3v v v v3 v 0v v + 80v v v + 40v v v v 50v v 0v v v v4 30v v v 4 + 6v 3 v v v v v 3 9 v v 80 3 v v 30 9 v v v 56 3 v v v v v 64 3 v v v v v + 0v v 4 + 4v v v v v v v v v 3 3 v v v 8 3 v v v 0v v 4 v v v 0v v v 4 6vv v 5 + 0v v v 4 + 0v v v 4 + 0v v v 4 v v 5 4v v v v 6 9 v v v v v v v v v v v v v v v 4 + v v v v v v 5 3

33 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, the derivatives v, v, remain in place All other derivatives are eliminated from the coefficients where they are alien 7 Continuum limit of lattice GD4 In order to get a truly multicomponent pluri-lagrangian system we move to the next equation of the GD hierarchy, with N = 4 In its 3-component form the discrete GD4 equation reads + U λ λ U V + + U λ λ U W λ V + λ V + λ W λ W V U V U + U W U W + V λ λ V W + W = V V = λ 4 λ 4 λ λ +U U + λ λ λ + λ + U λ λ U W W = λ 4 λ 4 λ λ +U U U U U + U, λ + λ U U + U U 4 U λ U + λ U, 5 + U λ λ U U λ U + λ U 6 Equations 5 and 6 do not just look similar, also their expansions are nearly identical in leading order: λ λ v = λ λ u u + λ λ u + λ λ u + Oλ + λ, λ λ w = λ λ u u λ λ u + λ λ u + Oλ + λ, where we used the constant c = in the Miwa correspondence 4 This gives us the ODE v w = u u + u, which integrates to v w = u + u 7 We have omitted the integration constant because higher order terms in the expansion force it to be zero Equation 7 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, V = X + Y, W = X Y Then in the continuum limit the variable y can be eliminated by Equation 7, which in the new variables reads y = u + u 8 33

34 One recognizes the Miura transformation [5] For the remaining two variables we find the hierarchy u = x, x = 4u x 8u x x, u 3 = 3 u + 4 u x, x 3 = 3u x x, 9 u 4 = 0, x 4 = 0, where, presumably, every fourth pair of equations is trivial The construction of the continuous GD4 hierarchy using pseudodifferential operators [6] yields the system φ = χ φ χ = χ + ψ φ φφ ψ = ψ φ φφ χφ as leading order equations They are related to the leading order equations of the hierarchy 9 by the transformation φ = 4u, χ = x + 4u 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 7 7 Pluri-Lagrangian structure For N = 4 we have the Lagrangian LU, V, W, U, V, W, U, V, W, U, V, W, λ, λ = λ 4 λ 4 log U + U λ λ λ 3 + λ λ + λ λ + λ 3 U U + λ + + λ λ λ U U U + U U V λ λ U + U U V λ V + λ V λ λ + U W V V + U U U + λ U λ U λ λ λ λ 30 As before, we make the change of variables V = X + Y, W = X Y 34