Mats Vermeeren
TU Berlin

Continuum limits of integrable lattice equations
and their variational structure
Multidimensionally consistent lattice equations

Integrable hierarchies of PDEs

Quad equation: Q(U, U1, U2, U12, λ1, λ2) = 0,invariant under symmetries of the square, affine ineach of the entries U, U1, U2, U12.

Hierarchy of commuting PDEs:
ut2 = f2(u, ux, uxx, . . .),ut3 = f3(u, ux, uxx, . . .), . . .

U12U2
λ2λ1U

Subscripts of U denote lattice shifts, λ1 and λ2 arelattice parameters.

U1

Integrable if consistent around the cube: initial dataU, U1, U2, U3 lead to a well-defined value of U123.
Then it can be imposed on a higher-dimensional
Nlattice Z .

A typical example is the potential KdV hierarchy
2ut3 = 3ux + uxxx,
32ut5 = 10ux + 5uxx + 10uxuxxx + uxxxxx, . . .

U123

U23
U3

U13
U2

λ3
U

By commutativity, we can consider a single fieldNu(x, t2, . . . , tN ) on higher-dimensional multi-time R .

U12λ2
λ1

U1

Lagrangian structure: discrete

Lagrangian structure: continuous
A field u : RN → R is a solution to thepluri-Lagrangian problem for the 2-form
XL[u] =Lij[u] dti ∧ dtj

A field U : ZN → R is a solution to thepluri-Lagrangian (or Lagrangian multiform)
problem forL(ij) = L(U, Ui, Uj, Uij, λi, λj)
if the action
XL()

i,j

if the actionZΓ∈ ΓL[u]Nis critical on all smooth surfaces Γ in R .

is critical on all discrete surfaces Γ in ZN .

Embedding the lattice into continuous multi-time: Miwa shifts
NN
Skew embedding of the mesh Z into multi-time R .

Correspondence between discrete and continuous fields:
U = U (n)= u(t1, t2, . . . , tN )
N2(−1)2
Ui = U (n + ei) = u t1 − 2λi, t2 + λi , . . . , tN + N λNiDiscrete U is a sampling of the continuous u.
We write the quad equation in terms of u, expand in λ1,λ2,and equate all coefficients of the power series to 0.
,→ Whole hierarchy from single difference equation.

[Miwa. On Hirota’s difference equations. Proc. Japan Acad. A. 1982]

In the leading order, we only see t1-derivatives of u, but we want to obtain PDEs.

→Leading order cancellation required.

Continuum limit of a Lagrangian
Z

Using Miwa shifts, turn discrete L(12) into power series
Ldisc([u], λ1, λ2). The action for Ldisc is still a sum.

Then

With the Euler-Maclaurin formula we turn the action sum into
an integral, with integrand∞XBiBj i jLMiwa([u], λ1, λ2) =∂1∂2Ldisc([u], λ1, λ2),i!j!

where η1 and η2 are the 1-forms dual to the Miwa shifts.

where the differential operators are ∂k =Pj (−1)

 12LMiwa([u(t)], λ1, λ2) η1 ∧ η2,

Pluri-Lagrangian 2-form for the continuum limit hierarchy:
XX
L=
LMiwa([u], λi, λj) ηi ∧ ηj =Lij[u] dti ∧ dtj,i<ji,j=0

j
j+1 2λk dL(12) =i<jwhere LMiwa([u], λ1, λ2) =.
j dt
jP∞i,j=1i+j(−1) 4ji
λ1 λ2i j Lij [u].

ExampleLattice potential KdV equation:−1−1λ1 + λ2 + U1,2 − U−1−1λ 2 − λ 1 + U2 − U1

−2−2= λ2 − λ1Miwa shifts

X−−−−→i jfi,j[u] λ1λ2 = 0fi,j =0−−→i,j≥0Lagrangian:1−1−1−1−1U − Ui,j − λi − λjUi − Uj + λi − λj
L=2 + λi−2 − λj−2 log 1 +Ui −Uj

λi−1 −λj−1.

Miwa shifts−−−−−→EM formulaV. Continuum limits of pluri-Lagrangian systems. arXiv:1706.06830

u2 = 0,
2+u
u=3u111
 31u4 = 0,


3 + 5u2 + 10u u

u=10u

51 111 + u11111,111
· · ·where ui = uti .

3−u uL=−
2u+u1u3
1,311111

4 + 10u u2 − u2 + u u


L
=−5u
1 51 11
 1,5111
1
L3,5 = 6u51 − 15u21u211 + 20u31u111 − 10u31u3 + 7u211u111 + 6u1u2111 − 12u1u11u1111

2u2 u − 10u u2u − u2
+3u+20uuu−
5uu+3u5311111
1 11 13
1 111 311111111
+u111u11111 − 2u111u113 + u1u115 + 2u1111u13 − u11u15 − u11111u3 + u111u5•V. Continuum Limits of Variational Sytems. PhD Thesis, 2018.

V. A variational perspective on continuum limits of ABS and lattice GD equations. arXiv:1811.01855•

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