Pith. sign in

REVIEW 3 major objections 5 minor 25 references

Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The ABS quad equations are the Euler–Lagrange equations of a discrete Lagrangian 2-form with auxiliary integer fields, making them variational in a literal sense.

desk verdict A real step forward in the variational theory of ABS equations, but the claim that 'quad equations are variational' only holds under a surface-dependent weakening of the multiform principle that the paper itself makes explicit. read the letter →

arxiv 2501.13012 v3 pith:ET4E35QF submitted 2025-01-22 nlin.SI math-phmath.MP

Jacob J. Richardson , Mats Vermeeren This is my paper · ORCID
classification nlin.SImath-phmath.MP MSC 39A3637J7037J06
keywords discreteLagrangianmultiformsABSequationsquadthree-legformbranchcutsinteger-valuedfieldsvariationalprinciplesmultidimensionalconsistency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that every quad equation on the ABS list (except Q4, which is left open) is genuinely variational: the equations are the Euler–Lagrange equations of a discrete Lagrangian 2-form, not merely consequences of one. Earlier Lagrangian multiforms for these equations used a three-point 'triangle' stencil and produced corner equations weaker than the quad equations; the paper shows that a four-point 'trident' stencil fixes this. Because converting the multiplicative three-leg form of most ABS equations into an additive one requires logarithms, branch cuts introduce ambiguities of multiples of $2\pi i$, and the paper's key move is to add integer-valued fields $\Theta$ and $\Xi$ to the action to absorb those ambiguities. With these fields, the corner equations are equivalent to the quad equations in three-leg form, and the action around an elementary cube vanishes on solutions for H1, H2, Q1, Q2, and A1, and is a multiple of $4\pi^2$ for H3, A2, and Q3. A sympathetic reader would care because this overturns a long-standing caveat in discrete integrable systems and provides a variational principle that captures the ABS equations exactly.

What carries the argument

The object that carries the argument is the trident Lagrangian, a four-point discrete Lagrangian arranged like a three-legged fork: one leg along direction $i$, one along direction $j$, and a diagonal leg connecting the base vertex to the opposite corner, with leg functions $\psi$ and $\phi$ arising as derivatives of potentials $L$ and $\Lambda$. Around a cube, six such Lagrangians are summed with signs to form the action $S_{\Theta,\Xi}$, which is extended by vertex terms $2\pi i\Theta U$ and direction terms $2\pi i\Xi_i A_i$. The integer fields do the load-bearing work: $\Theta$ makes the three-leg form exactly equivalent to the multi-affine quad equation despite logarithm branch cuts, and $\Xi$ cancels the branch jumps so that the gradient of the action with respect to both fields and parameters vanishes on solutions. The closure proof then works by deforming a given solution to a trivial one along a one-parameter family, using Lemma 3.3 (a 2-form version of the spectrality property, proved via biquadratic identities) to choose $\Xi$, and Lemma 3.4 to control jumps when branch cuts are crossed.

What would settle it

For H3, A2, or Q3, take a one-parameter family of solutions to the quad equations that crosses a branch cut of the dilogarithm, and compute the extended action $S_{\Theta,\Xi}$ around an elementary cube; if the jump is ever a non-integer multiple of $4\pi^2$, Theorem 3.5's closure claim is false. A direct search for a closed discrete surface whose total action equals, for instance, $2\pi^2$ would also refute the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the multi-affine ABS quad equations are equivalent to the corner equations of a discrete 2-form action $S_{\Theta,\Xi}$ built from the trident Lagrangian $\mathcal{L}(U,U_i,U_j,U_{ij},A_i,A_j)=L(U,U_i,A_i)-L(U,U_j,A_j)-\Lambda(U,U_{ij},A_i-A_j)$. The integer fields $\Theta$ (one per lattice site) turn the additive three-leg equations, which otherwise hold only up to $2\pi i$, into exact equivalents of the multiplicative quad equations; the fields $\Xi$ (one per lattice direction) absorb branch-cut jumps in the logarithm and dilogarithm terms, so that the action's derivatives with respect to the lattice parameters vanish. Theorem 3.1 states that the corner equations of $S_{\Theta,\Xi}$ are precisely the quad equations and tetrahedron equations in three-leg form, and Theorem 3.5 states that on solutions the action around an elementary cube equals $0$ for H1, H2, Q1, Q2, and A1, and equals $4k\pi^2$ with $k\in\mathbb{Z}$ for H3, A2, and Q3. This makes the quad equations of the ABS list variational in a literal sense, contrary to the statement in earlier work that they are not.

Load-bearing premise

The whole construction depends on allowing the integer fields $\Theta$ and $\Xi$ to be chosen anew for each discrete surface; if a single global assignment on $\mathbb{Z}^N$ were required, the variational description would fail, as the paper's own H2 example on two adjacent cubes demonstrates.

Editorial extensions

If this is right

  • The earlier caveat that 'quad equations are not variational' is no longer true: the trident 2-form gives corner equations equivalent to the ABS quad equations themselves.
  • For H1, H2, Q1, Q2, and A1, the Lagrangian 2-form is closed on solutions in the strong sense that the action over every elementary cube is zero.
  • For H3, A2, and Q3, closure holds modulo $4\pi^2$, and numerical evidence in the paper shows nonzero multiples occur, so the variational structure of these equations is best described as a pluri-Lagrangian system.
  • The same branch-tracking technique provides a concrete route toward a closure relation for Q4, where the paper expects the $4\pi^2$ multiple to be replaced by quantities related to half-periods of the underlying elliptic curve.
  • Definition 3.7 proposes the appropriate formulation of the Lagrangian multiform principle in the presence of integer fields: the fields may depend on the chosen discrete surface, even though no global assignment exists on the whole lattice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the construction extends to Q4, the entire ABS list would sit on a uniform variational foundation, and the $4\pi^2$ obstruction for H3, A2, and Q3 might be interpretable as a topological term tied to elliptic periods.
  • The surface-dependent integer fields behave like a discrete analogue of a connection: the action is not a single-valued function on field space but a section of a bundle with $\mathbb{Z}$-valued holonomy, which could give the integer fields a geometric meaning the paper leaves open.
  • The same 'add integer fields to absorb branch ambiguities' recipe may apply to other discrete integrable systems whose Lagrangians involve logarithms or dilogarithms, such as star-triangle relations, where similar multiples of $2\pi i$ appear.
  • A testable consequence of Definition 3.7 is that the variational principle and the quad equations remain equivalent on arbitrary discrete surfaces; checking this on surfaces with many cubes, where multiple $\Theta$ values meet at interior vertices, would stress-test Proposition 3.8.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents new discrete Lagrangian multiforms for the ABS list of quad equations, based on a four-point (trident) Lagrangian instead of the usual three-point (triangle) Lagrangian. The authors show that the corner equations of the trident action are equivalent to the quad equations, provided one adds integer-valued fields Θ and Ξ to account for branch choices of logarithms and dilogarithms. They also revisit the closure property, give counterexamples to existing closure proofs, and prove exact closure for H1, H2, Q1, Q2, A1 and closure modulo 4π² for H3, A2, Q3, using a deformation argument that tracks branch jumps. The final section extends the construction to arbitrary surfaces by allowing the integer fields to depend on the surface, and states an open problem for Q4.

Significance. If accepted as a generalized variational principle, this is a substantial contribution: it provides a variational formulation equivalent to the ABS quad equations, clarifying a long-standing issue in discrete Lagrangian multiform theory, and it draws attention to branch-cut subtleties that earlier works ignored. The paper is honest about the limits of its construction, including the absence of exact closure for H3, A2, Q3 and the surface-dependence of the integer fields. Strengths include explicit counterexamples, a deformation proof of closure, and the availability of SageMath verification code at a Zenodo repository. The central derivation in Section 3.2 is clean and the corner-equation computation is straightforward once the trident action is written out. The main risk is definitional: whether the surface-dependent integer fields in Definition 3.7 really produce a Lagrangian multiform in the established sense, and whether the branch-jump classification in Lemma 3.4 is complete.

major comments (3)
  1. [Section 3.5, Definition 3.7] The proposed Lagrangian multiform principle is weaker than the standard one: the integer fields Θ and Ξ are allowed to depend on the chosen discrete surface Γ, so the action S_{Θ,Ξ}^Γ is not a single fixed functional on Z^N. The H2 example in equations (3.6) and (3.7) shows that no global assignment of Θ and Ξ exists even for two adjacent cubes, so the standard interpretation of a Lagrangian multiform as a fixed 2-form on the lattice is not realized. The claim in the Introduction and Section 3.2 that 'quad equations are variational' should therefore be explicitly qualified as 'variational in this surface-dependent generalized sense'; without that qualification, the statement is misleading under the usual definition.
  2. [Lemma 3.4, proof] The proof of Lemma 3.4 classifies branch jumps into three term types and then asserts that all terms in the actions are of these forms. However, the actions in Appendix A contain many logarithms and dilogarithms with different arguments and sign patterns, and the proof does not provide a systematic check that each equation's action term satisfies the claimed coefficient relations (for example, that a jump of 2πi in a logarithm of a combination of fields is always multiplied by the same combination of fields and parameters, or that the dilogarithm terms always appear in the specific pairing claimed). Since Lemma 3.4 is load-bearing for Theorem 3.5, please add a per-equation verification table or a general argument that covers every logarithmic/dilogarithmic term in the listed Lagrangians.
  3. [Proof of Theorem 3.5 for H3, Q3, A2] The proof for H3, Q3, A2 relies on a generic assumption that w_i, w_j, w_k are distinct, with non-generic cases handled by 'a small perturbation of the original solution'. This is not justified: it is not shown that the value of S_{Θ,Ξ}(U,A,Θ,Ξ) modulo 4π² is continuous under such perturbations, especially since Θ(t) and Ξ(t) are piecewise-constant integer fields that can change when branch cuts are crossed. The proof should either give a direct argument for the non-generic cases or prove that the mod-4π² value is locally constant on the solution manifold in a way that survives the perturbation limit.
minor comments (5)
  1. [Theorem 3.5 statement] The phrase 'Let either S = S_{Θ,Ξ} or S = S_{Θ,Ξ}' is confusing because the two symbols are identical in the printed text; presumably the second should be S_{Ξ} (or the statement should specify whether Θ is included). Please correct the notation.
  2. [Lemma 3.3 statement] The first sentence of Lemma 3.3 repeats 'there exists a choice of integers Ξ_i, Ξ_j, Ξ_k ∈ Z such that ∂S_Ξ/∂α_i = ∂S_{Θ,Ξ}/∂α_i = 0' twice, with the second occurrence presumably intended to refer to S_{Ξ} or to an analogous condition for S_{Θ,Ξ} without the Θ terms; please rephrase to state clearly which action is meant in each case.
  3. [Example H3, part 3, numerical values] The numerical solution is described as 'a (non-unique) solution', but no numerical method or precision is given. Since the paper already points to the SageMath code, a brief sentence on how the numbers were computed (e.g., by solving the quad equations with a root-finding algorithm) would improve reproducibility.
  4. [The deformation family in the proof for H1, H2, Q1, Q2, A1] The claim that lim_{t→0} S(V(t), Θ, Ξ) = 0 is asserted to follow from 'elementary calculus', but the dilogarithm terms in H2 and Q1δ=1, Q2 involve limits of terms like x log x as x→0; a short justification of the vanishing of such limits would make the proof self-contained.
  5. [General typographical issues] There are several places where the text says 'A1δ=1' or 'Q1δ=1' etc., but the subscript formatting is inconsistent (e.g., 'Q1δ=1' versus 'Q1_{δ=1}'). These do not affect the mathematics but should be fixed in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the corner-equation equivalence and closure theorems are derived from the constructed action, not assumed.

full rationale

The paper's central construction is a standard inverse-problem-of-calculus-of-variations argument: the leg functions ψ and φ are taken from the ABS three-leg forms, and L and Λ are chosen as their antiderivatives (equations (2.16)–(2.17)). Differentiating the action then produces the three-leg expressions by direct calculus, and Proposition 2.3 independently establishes the equivalence between the multi-affine quad equations and those three-leg forms modulo 2πi. The integer fields Θ and Ξ are not fitted to force the claimed conclusions; they are introduced to absorb branch-of-logarithm ambiguities, and the theorems are then proved rather than assumed. In Theorem 3.1, the corner equations are shown to be the three-leg quad and tetrahedron equations, and the equivalence to the ABS equations follows from Proposition 2.3, not from the definition of the action. In Theorem 3.5, closure is proved by showing that the gradient of SΘ,Ξ vanishes on solutions (using the corner equations and Lemma 3.3), then deforming a solution to a trivial solution where S→0, and tracking branch jumps in Lemma 3.4. The integers Ξ are chosen to satisfy ∂S/∂A=0, not to set S=0 directly, so the conclusion S=0 is a genuine result rather than a fitted prediction. The paper is also transparent about its limitations: for H3, A2, and Q3 closure holds only modulo 4π², and Section 3.5 explicitly demonstrates that no global assignment of the integer fields exists for the H2 example, motivating the surface-dependent Definition 3.7. That weakening of the standard Lagrangian-multiform principle is a substantive interpretive and correctness concern, but it is not circularity: the paper does not define the variational principle in terms of the quad equations, and its derivation chain does not reduce to its own inputs by construction. The few self-references, such as [17] and [23], are not load-bearing evidence for the main theorems. The derivation is self-contained, and no circular step can be exhibited by quoting an equation that is equivalent to another by definition or by a fitted parameter renamed as a prediction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The construction rests on the ABS classification's structural facts (three-leg forms, biquadratic identities, tetrahedron representation) and on two ad hoc devices introduced in this paper: solution-dependent integer fields and the mod-4π² closure condition. These devices are what convert a known planar Lagrangian into a multiform with the desired corner equations.

free parameters (2)
  • Integer fields Θ(n) at each lattice site = Solution-dependent integers; H2 example: Θ = -1, Θ_i = 1, Θ_k = 1, Θ_ij = 1, Θ_jk = -1, Θ_ki = -1, Θ_ijk = 1
    Introduced in equation (2.22) and the cube action SΘ,Ξ to absorb the 2πi ambiguities that arise when exponentiating additive three-leg forms; the corner equations hold only up to these integers, so the equivalence with ABS quad equations depends on their existence.
  • Integer fields Ξ_i per lattice direction = Solution-dependent integers; H1 example gives Ξ_i = 0, Ξ_j = -1, Ξ_k = 1; the adjacent-cube H2 example gives…
    Introduced in Section 3.1 to cancel branch jumps in the action and enforce (almost-)closure; their values are determined by ∂SΞ/∂A_i = 0, and Section 3.5 shows they cannot be fixed globally.
assumptions (6)
  • standard math Biquadratic identities and the relation ∂L/∂A ≡ log h + ... mod 2πi from [3, Lemma 3] and [1, Proposition 15]
    Used in Lemma 3.3 to prove existence of Ξ fields; if the mod-2πi identities fail for the paper's non-symmetric L and Λ, the closure proof collapses.
  • domain assumption The tetrahedron equation T=0 can be represented as a quad equation of type Q evaluated on a tetrahedron stencil
    Invoked in Section 2.1 and in Lemma 3.3; this is a structural fact of the ABS classification.
  • domain assumption Choice of principal branches for log and dilogarithm with specified branch cuts
    The paper defines L and Λ with logarithms and dilogarithms; the counterexamples and closure results are relative to this branch convention, as discussed in Section 3.4.
  • ad hoc to paper Definition 3.7: integer fields Θ and Ξ may depend on the chosen discrete surface
    This weakens the standard Lagrangian multiform principle; without it, the H2 example in Section 3.5 shows adjacent cubes require conflicting integer fields, so no single extended action exists on Z^N.
  • ad hoc to paper The closure requirement is relaxed to S ≡ 0 mod 4π² for H3, A2, and Q3
    Theorem 3.5 and Section 3.4 establish only mod-4π² closure for these equations, and numerical evidence shows nonzero multiples occur; exact closure is open.
  • ad hoc to paper Generic solution assumption wi, wj, wk distinct in the proof for H3, A2, Q3, with non-generic cases reached by perturbation
    Used in the proof of Theorem 3.5; the perturbation argument relies on Lemma 3.4's branch-jump classification.
invented entities (2)
  • Integer-valued field Θ(n) at each lattice site
    purpose: Absorb multiples of 2πi so that three-leg corner equations are equivalent to the multi-affine ABS quad equations
    Introduced in equations (2.22) and (3.2); no physical or geometric interpretation is provided, and values are fixed per solution and per surface.
  • Integer-valued field Ξ_i per lattice direction
    purpose: Cancel branch-cut jumps in the action and enforce (almost-)closure of the discrete 2-form
    Introduced in Section 3.1; cannot be assigned consistently to adjacent cubes (Section 3.5, H2 example), so they are auxiliary bookkeeping fields rather than fixed structure.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations." pith.science (2026). https://pith.science/paper/ET4E35QF

@misc{pith2026250113012,
  author       = {Pith},
  title        = {Pith review of: Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ET4E35QF}},
  note         = {Machine review of arXiv:2501.13012}
}
read the original abstract

Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums.

Figures

Figures reproduced from arXiv: 2501.13012 by the authors.

Figure 1
Figure 1. Multidimensional consistency demands that all three of these routes to calculate uijk from initial values u, ui , uj , uk produce the same value. The ABS list is given in Appendix A. By convention, its members are denoted H1–H3, A1– A2, and Q1–Q4. Some of them depend on a parameter δ, and we use a subscript on the equation name to indicate whether or not this parameter is zero. Each equation Q(u, ui , uj , uij , αi … view at source ↗
Figure 2
Figure 2. Graphical representation of the four orientations of a three-leg form, where the colour reflects the sign of the leg function. We denote the additive three-leg expression, based at the vertex u, by Q (u) ij := ψ(u, ui , αi) − ψ(u, uj , αj ) − ϕ(u, uij , αi − αj ). (2.5) We can also consider three-leg forms based at the other three vertices of the square: Q (ui) ij := ψ(ui , uij , αj ) − ψ(ui , u, αi) − ϕ(ui , uj , α… view at source ↗
Figure 3
Figure 3. Tetrahedron equation from three quad equations. Example: H2, part 1. Consider equation H2, for which Qij = (u − uij )(ui − uj ) − (αi − αj )(u + ui + uij + uj ) − α 2 i + α 2 j . Its multiplicative three-leg form is given by αi − αj + u − uij −αi + αj + u − uij · αi + u + ui αj + u + uj = 1. Hence, the additive three-leg form is given by log(αi + u + ui) − log(αj + u + uj ) − log  −αi + αj + u − uij αi − αj + u − u… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (a) The stencil of the trident Lagrangian. (b) The discrete Euler–Lagrange equation involves three-leg structures on two squares. U Ui Uj L(U, Ui , Ai) − L(U, Uj , Aj ) −Λ(Ui , Uj , Ai − Aj ) (a) • • • U−i,j Uj Ui U−j Ui,−j U−i U (b) • • • • • • • [PITH_FULL_IMAGE:fig…
Figure 5
Figure 5. Figure 5: (a) The stencil of the triangle Lagrangian. (b) The discrete Euler–Lagrange equation involves three-leg structures on two squares. Alternatively, the functions L and Λ can be combined into a 3-point Lagrangian [12] L (U, Ui , Uj , Ai , Aj ) := L(U, Ui , Ai) − L(U, Uj ,…
Figure 6
Figure 6. Figure 6: (a) The leg structure of a single Lagrangian L (Uk, Uki, Ujk, Uijk, Ai , Aj ). (b) The leg struc￾ture for the action on the cube of the trident 2-form L . (in their three-leg form). The action over an elementary cube of L , interpreted as a 2-form, is illustrated in […
Figure 7
Figure 7. Figure 7: (a) The leg structure of a single Lagrangian L (Uk, Uki, Ujk, Ai , Aj ). (b) The leg structure for the action on an elementary cube of the triangle 2-form L sits on an octahedral stencil. For equations other than H1, Q1δ=0 and A1δ=0, solutions to the quad equations may…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 24 canonical work pages

  1. [1]

    The consistency approach, Comm

    Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys.233 (2003), 513–543, arXiv:nlin.SI/0202024

    work page arXiv 2003

  2. [2]

    Bobenko A.I., G¨ unther F., On discrete integrable equations with convex variational principles, Lett. Math. Phys. 102 (2012), 181–202, arXiv:1111.6273

    work page Pith review arXiv 2012

  3. [3]

    Bobenko A.I., Suris Yu.B., On the Lagrangian structure of integrable quad-equations, Lett. Math. Phys.92 (2010), 17–31, arXiv:0912.2464

    work page Pith review arXiv 2010

  4. [4]

    Bobenko A.I., Suris Yu.B., Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems, Comm. Math. Phys.336 (2015), 199–215, arXiv:1403.2876. 30 J.J. Richardson and M. Vermeeren

    work page Pith review arXiv 2015

  5. [5]

    Boll R., Petrera M., Suris Yu.B., What is integrability of discrete variational systems?, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.470 (2014), 20130550, 15 pages, arXiv:1307.0523

    work page Pith review arXiv 2014

  6. [6]

    Boll R., Petrera M., Suris Yu.B., On integrability of discrete variational systems: octahedron relations, Int. Math. Res. Not.2016 (2016), 645–668, arXiv:1406.0741

    work page Pith review arXiv 2016

  7. [7]

    Caudrelier V., Nijhoff F., Sleigh D., Vermeeren M., Lagrangian multiforms on Lie groups and non-commuting flows, J. Geom. Phys.187 (2023), 104807, 35 pages, arXiv:2204.09663

    work page Pith review arXiv 2023

  8. [8]

    Caudrelier V., Stoppato M., Vicedo B., Classical Yang–Baxter equation, Lagrangian multiforms and ul- tralocal integrable hierarchies, Comm. Math. Phys.405 (2024), 12, 67 pages, arXiv:2201.08286

    work page Pith review arXiv 2024

Show all 25 references
  1. [9]

    Math., Cambridge University Press, Cambridge, 2016

    Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2016

    2016

  2. [10]

    Kels A.P., Interaction-round-a-face and consistency-around-a-face-centered-cube, J. Math. Phys.62 (2021), 033509, 36 pages, arXiv:2003.08883

    2021 arXiv

  3. [11]

    D 448 (2023), 133723, 23 pages, arXiv:1910.03562

    Kels A.P., Two-component Yang–Baxter maps and star-triangle relations, Phys. D 448 (2023), 133723, 23 pages, arXiv:1910.03562

    2023 arXiv

  4. [12]

    Lobb S.B., Nijhoff F.W., Lagrangian multiforms and multidimensional consistency, J. Phys. A42 (2009), 454013, 18 pages, arXiv:0903.4086

    2009 arXiv

  5. [13]

    Lobb S.B., Nijhoff F.W., A variational principle for discrete integrable systems, SIGMA 14 (2018), 041, 18 pages, arXiv:1312.1440

    2018 arXiv

  6. [14]

    Nijhoff F.W., Lagrangian 3-form structure for the Darboux system and the KP hierarchy, Lett. Math. Phys. 113 (2023), 27, 19 pages, arXiv:2206.14338

    2023 arXiv

  7. [15]

    Nonlinear Math

    Nijhoff F.W., Lagrangian multiform structure of discrete and semi-discrete KP systems, Open Commun. Nonlinear Math. Phys.4 (2024), OCNMP Conference, 91–115, arXiv:2406.13423

    2024 arXiv

  8. [16]

    Non- linear Math

    Petrera M., Suris Yu.B., Variational symmetries and pluri-Lagrangian systems in classical mechanics,J. Non- linear Math. Phys.24 (2017), suppl. 1, 121–145, arXiv:1710.01526

    2017 arXiv

  9. [17]

    Richardson J.J., Vermeeren M., Discrete Lagrangian multiforms for quad equations II: Tetrahedron and octahedron equations, SIGMA 21 (2025), 059, 27 pages, arXiv:2403.16845

    2025 arXiv

  10. [18]

    Sleigh D., Nijhoff F.W., Caudrelier V., Lagrangian multiforms for Kadomtsev–Petviashvili (KP) and the Gelfand–Dickey hierarchy, Int. Math. Res. Not.2023 (2023), 1420–1460, arXiv:2011.04543

    2023 arXiv

  11. [19]

    Sleigh D., Vermeeren M., Semi-discrete Lagrangian 2-forms and the Toda hierarchy, J. Phys. A55 (2022), 475204, 24 pages, arXiv:2204.13063

    2022 arXiv

  12. [20]

    Suris Yu.B., Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms, J. Geom. Mech.5 (2013), 365–379, arXiv:1212.3314

    2013 arXiv

  13. [21]

    Suris Yu.B., Vermeeren M., On the Lagrangian structure of integrable hierarchies, in Advances in Discrete Differential Geometry, Springer, Berlin, 2016, 347–378, arXiv:1510.03724

    2016 arXiv

  14. [22]

    The Sage Developers, SageMath, the Sage Mathematics Software System, Version 9.1 of 2020-05-21, aviable at https://www.sagemath.org

    2020

  15. [23]

    Vermeeren M., SageMath code ABS-multiforms: v1.1, 2025, aviable at https://zenodo.org/doi/10.5281/ zenodo.15652765

    2025

  16. [24]

    Vermeeren M., A variational perspective on continuum limits of ABS and lattice GD equations, SIGMA 15 (2019), 044, 35 pages, arXiv:1811.01855

    2019 arXiv

  17. [25]

    Xenitidis P., Nijhoff F., Lobb S., On the Lagrangian formulation of multidimensionally consistent systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.467 (2011), 3295–3317, arXiv:1008.1952

    2011 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.