Pith. sign in

REVIEW 3 major objections 3 minor 51 references

Arithmetic selection rules in dispersionless Hamiltonian systems

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper shows that for the Hamiltonian $H=\int \pi^2 \phi'^3$, the Liouville integrability conditions reduce to the negative Pell equation $x^2-3y^2=-2$, whose infinite solution family yields infinitely many mutually commuting monomial…

desk verdict Solid, checkable algebra with one real omission: the Pell charges are shown to commute, but functional independence is never proved, so the Liouville claim needs a small patch. read the letter →

arxiv 2608.11179 v1 pith:3G3XGD5M submitted 2026-08-11 nlin.SI hep-thmath-phmath.MP

Fatma Aydogmus , Mustafa Mullahasanoglu This is my paper · ORCID
classification nlin.SIhep-thmath-phmath.MP MSC 37K1037K0511D0935Q53
keywords HamiltoniansystemsLiouvilleintegrabilitydispersionlessmonomialchargesnegativePellequationMotzkinpolynomialsderivativenonlinearSchrödingerBurgers
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

Liouville integrability for monomial charge densities $I_{i,j}=\int\pi^i\phi'^j$ in a 1+1-dimensional Hamiltonian field theory is shown to impose two arithmetic selection rules on the exponent pairs $(i,j)$. For the specific Hamiltonian $H=\int\pi^2\phi'^3$, the conservation selection rule reduces to the negative Pell equation $x^2-3y^2=-2$ with $x=2j-1$, $y=2i-1$, and its infinite solution family is shown to satisfy the involution rule as well, yielding infinitely many mutually commuting conserved charges. The paper also establishes that the Motzkin model is a dispersionless limit of the Levi system, that the binomial model is the dispersionless derivative nonlinear Schrödinger equation (Kaup–Newell system), and that the binomial composite field obeys an inviscid Burgers equation. If these claims hold, the result is a new dispersionless Liouville integrable system and a combinatorial-to-dispersive dictionary.

What carries the argument

The central object is the family of monomial charge densities $I_{i,j}=\int \pi^i \phi'^j$, with Hamiltonian $H=I_{r,s}=\int \pi^r \phi'^s$. The argument's engine is the pair of selection rules (5.5) and (5.7), obtained by demanding that $\dot I_{i,j}$ and $\{I_{n,m},I_{k,l}\}$ be spatial total derivatives. For the Hamiltonian exponents $(r,s)=(2,3)$, the conservation rule becomes the negative Pell equation $x^2-3y^2=-2$, and the involution rule becomes the requirement that differences of two such Pell equations vanish; the infinite solution family $x_n+y_n\sqrt3=(1+\sqrt3)(2+\sqrt3)^n$ supplies the allowed exponent pairs.

What would settle it

Compute the variational derivatives of the first several Pell charges $I_{i,j}$ at a generic field configuration and test their linear independence; if any finite subset is functionally dependent, the infinite hierarchy cannot establish Liouville integrability under the paper's own definition.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the two conditions of Liouville integrability for the monomial class $I_{i,j}=\int\pi^i\phi'^j$ reduce to arithmetic selection rules on the exponents. For the Hamiltonian exponents $(r,s)=(2,3)$, the conservation rule becomes the negative Pell equation $x^2-3y^2=-2$, and the involution rule becomes the statement that a difference of two such Pell equations vanishes; consequently every two members of the Pell solution family are in involution. The paper interprets the infinite solution family $(i,j)=(1,1),(2,3),(6,10),(21,36),\ldots$ as an infinite set of mutually commuting conserved charges, and therefore asserts that the resulting equations of motion define a novel dispersionless class of integrable systems. It further demonstrates that the previously studied Motzkin and binomial models are dispersionless reductions of known integrable systems: the Levi system and the DNLS/Kaup–Newell system, respectively, with the binomial composite field satisfying a Burgers-type hierarchy.

Load-bearing premise

The argument collapses if the infinite family of commuting Pell charges turns out to be functionally dependent, because the paper's own Liouville integrability criterion requires an infinite set of functionally independent charges and that independence is assumed rather than proved.

Editorial extensions

If this is right

  • For $H=\int\pi^2\phi'^3$, the Pell solution family $(1,1),(2,3),(6,10),(21,36),\ldots$ gives an infinite set of mutually commuting conserved charges; if these are functionally independent, the model is a new dispersionless Liouville integrable system.
  • The Motzkin model coincides with the dispersionless Levi system, so its polynomial charge hierarchy is a dispersionless reduction of a known integrable dispersive system.
  • The binomial model is the dispersionless DNLS (Kaup–Newell) system, and adding a constant shift $k$ preserves integrability while modifying the lower-power flux coefficient from $ab$ to $ab-ck$.
  • The composite field of the binomial model satisfies an inviscid Burgers equation, and the higher-order charges generate generalized Burgers-type conservation laws for the two-component system.
  • The selection-rule method turns the search for Liouville-integrable monomial models into a Diophantine classification problem over the exponent pairs $(r,s)$.

Reading between the lines

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

  • Because the Pell exponents grow like $(2+\sqrt3)^n$, the hierarchy is sparse: the method excludes almost all exponent pairs, so an interesting open question (which the paper leaves implicit) is which other $(r,s)$ admit infinite Diophantine families.
  • The functional-independence gap is the main risk: if the first few Pell charges are functionally dependent, the 'new model' may be a disguised version of a known binomial-type hierarchy.
  • The Burgers/Cole–Hopf structure of the binomial model suggests a diffusion-regularized interpretation that the paper mentions but does not classify as a separate integrable hierarchy; testing the commuting flows numerically on a lattice would reveal how many genuinely independent conserved quantities survive.
  • The selection rules could be applied to other Hamiltonian exponent pairs as a systematic generator of candidate integrable monomial models, and dispersive deformations of the Pell model may connect it to the known NLS/DNLS network.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies monomial charge densities I_{i,j} = ∫ π^i φ'^j in a 1+1-dimensional Hamiltonian field theory with Hamiltonian H = I_{r,s}. It derives two arithmetic selection rules: one from the conservation condition dI_{i,j}/dt = 0 and one from the involution condition {I_{n,m}, I_{k,l}} = 0. For (r,s) = (2,3), the first rule reduces to the negative Pell equation (2j−1)^2 − 3(2i−1)^2 = −2, whose solutions (1,1), (2,3), (6,10), (21,36), ... yield an infinite family of commuting monomial charges. The paper also identifies the Motzkin model as the dispersionless limit of the Levi system and the binomial model as a dispersionless DNLS/Burgers-type system, with higher charges generating generalized Burgers conservation laws.

Significance. The core algebraic construction is sound: I independently re-derived the reduction of the conservation condition to j^2 − j = 3(i^2 − i), confirmed that the listed Pell pairs satisfy it, and verified that the involution condition (5.7) is consistent with a direct Poisson-bracket computation. If the missing functional-independence lemma is supplied, the Pell family provides a genuinely new dispersionless Liouville-integrable model with an explicit Diophantine hierarchy. The correspondences with the Levi and DNLS systems are useful and clearly presented. The paper is self-contained, with no fitted parameters and no circular reasoning. Its main deficits are a misprinted intermediate formula for the Poisson bracket in Eq. (5.6) and the absence of a functional-independence proof for the Pell charges.

major comments (3)
  1. [Section 5, Eq. (5.6)] The displayed formula for {I_{n,m}, I_{k,l}} is written as the x-derivative of a single product, so the integral would vanish identically and the second selection rule (5.7) would be unnecessary. A direct computation gives {I_{n,m}, I_{k,l}} = ∫ [ nk(l−m) π^{n+k−2} π' φ'^{m+l−1} + (nl(l−1) − km(m−1)) π^{n+k−1} φ'^{m+l−2} φ'' ] dx, which is not an exact derivative. After one integration by parts it equals ( C/(m+l−1) − A/(n+k−1) ) ∫ π^{n+k−1} ∂_x(φ'^{m+l−1}) dx, with A = nk(l−m) and C = nl(l−1) − km(m−1); the bracket vanishes iff A/(n+k−1) = C/(m+l−1), which is exactly condition (5.7). The intermediate expression (5.6) should be corrected to show this reduction rather than written as an identically vanishing total derivative.
  2. [Section 5.1, after Eq. (5.19)] The paper asserts that the Pell family of charges I_{i,j} defines a Liouville-integrable model, but it never proves that these charges are functionally independent, a requirement stated explicitly in Section 2. Without such a proof, an infinite set of commuting charges could in principle be redundant, and the model would not satisfy the paper's own integrability criterion. The missing lemma is elementary: for any finite subset, an identity ∑_a c_a δI_{i_a,j_a}/δπ = ∑_a c_a i_a π^{i_a−1} φ'^{j_a} = 0 forces all c_a = 0 because the monomials have distinct exponent pairs. Please add this argument (and the analogous statement for the Hamiltonian) to complete the Liouville-integrability claim.
  3. [Section 4, text after Eq. (4.5)] The sentence 'Together with the infinite conserved quantities in (4.1), this establishes the formal Liouville integrability of the binomial model' repeats the same omission. If the word 'formal' is intended to signal that functional independence is not being claimed, this should be stated explicitly; otherwise the independence of the charges I_n = ∫ P^n should be checked. The same linear-independence argument on the gradients applies and should be included or referenced.
minor comments (3)
  1. [Section 5.1, Eqs. (5.22)–(5.23)] The step from the second selection rule to Eq. (5.22) is asserted rather than shown; please display the substitutions m(m−1) = 3n(n−1) and l(l−1) = 3k(k−1) so the reader can verify the algebra without reconstructing it.
  2. [Section 3.1, Eqs. (3.9)–(3.14)] The correspondence with the Levi system should be phrased explicitly as an equation-level dispersionless limit, since the zero-curvature representation degenerates to an Abelian form and loses the spectral data; the text already acknowledges this, but the conclusion could be stated more carefully.
  3. [References] Reference 2 contains a typo: 'Hamltonian methods' should be 'Hamiltonian methods'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the Pell selection-rule derivation is self-contained algebra, and the only self-citation is non-load-bearing.

full rationale

The derivation chain is algebraically self-contained and does not reduce any claimed result to its inputs. The central Pell construction proceeds as follows: for H = I_{2,3}, the exact-derivative conservation condition on I_{i,j} yields the negative Pell equation (5.17); the involution condition (5.10) is then checked by substituting that Pell condition, giving (5.22) and (5.23). Since both members of any pair of charges in the family satisfy the same Pell relation, (5.23) becomes an identity, so the involution verification is a direct algebraic consequence rather than a circular step. No fitted constants are promoted to predictions, and no externally imposed uniqueness theorem is invoked. The only self-citation is to [8] for the binomial model, but Section 4 explicitly re-derives conservation (4.4) and involution (4.5), so the citation is not load-bearing. A genuine but non-circular correctness gap is worth flagging: Section 2 defines Liouville integrability as requiring functionally independent integrals, while Section 5.1 concludes integrability of the Pell family without proving functional independence of the infinite set of monomial charges. This is an omitted proof rather than a self-referential reduction, so it does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivation is analytical and self-contained. The only hand-chosen input is the exponent pair (r,s) = (2,3) in the Hamiltonian, which is selected to yield a negative Pell equation. The key unproved assumption is functional independence of the infinite tower of Pell-family charges. No new physical entities are introduced.

free parameters (1)
  • Hamiltonian exponent pair (r,s) = (2,3)
    Chosen by hand to produce the negative Pell selection rule; other choices give known models (2,1) and (2,2). Not fitted to data, but selected to demonstrate a new integrable case.
assumptions (5)
  • standard math Standard canonical Poisson bracket and Hamilton equations in 1+1 dimensions
    Section 2 sets the Hamiltonian framework assumed throughout the paper.
  • domain assumption Boundary terms vanish (periodic fields or rapid spatial decay)
    Section 2 explicitly assumes this to discard total derivatives in conservation proofs.
  • domain assumption A local conserved charge must have its time derivative expressible as a total x-derivative
    Section 5 uses this to derive the first selection rule (5.5); this is the core mechanism of the conservation test.
  • standard math The infinite solution set of the negative Pell equation x^2 - 3y^2 = -2 is generated by (1+sqrt(3))*(2+sqrt(3))^n
    Section 5.1 cites references [36,37] and uses this standard number-theory fact to enumerate the charge family.
  • ad hoc to paper The integrals of motion are functionally independent
    Section 2 states this as a requirement, but the Pell construction never verifies it. This is a gap in the Liouville integrability claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Arithmetic selection rules in dispersionless Hamiltonian systems." pith.science (2026). https://pith.science/paper/3G3XGD5M

@misc{pith2026260811179,
  author       = {Pith},
  title        = {Pith review of: Arithmetic selection rules in dispersionless Hamiltonian systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3G3XGD5M}},
  note         = {Machine review of arXiv:2608.11179}
}
read the original abstract

In this work, we derive selection rules imposed by Liouville integrability conditions for monomial charge densities with arbitrary powers. For a certain monomial Hamiltonian system, the selection rules reduce to a negative Pell equation, and its solutions generate an infinite set of integrals of motion that are mutually in involution. Furthermore, we study the correspondence between combinatorial polynomial sequences and Liouville integrable Hamiltonian field theories in 1+1 dimensions. We show that the Motzkin system coincides with the dispersionless limit of the Levi system, while the binomial system is equivalent to the dispersionless derivative nonlinear Schr\"odinger equation. Additionally, we show that the binomial Hamiltonian model admits a reduction to the inviscid Burgers equation and its higher-order charges generate generalized Burgers-type conservation laws.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 43 canonical work pages

  1. [1]

    V. I. Arnol’d, Mathematical methods of classical mechanics, vol. 60. Springer Science & Business Media, 2013

    work page 2013

  2. [2]

    L. D. Faddeev and L. A. Takhtajan, Hamltonian methods in the theory of solitons. Springer Berlin, Heidelberg, 2007

    work page 2007

  3. [3]

    Introduction to classical and quantum integrability,

    A. L. Retore, “Introduction to classical and quantum integrability,” J. Phys. A55(2022) no. 17, 173001,arXiv:2109.14280 [hep-th]

    arXiv 2022

  4. [4]

    Modave Lectures on Classical Integrability in 2d Field Theories,

    S. Driezen, “Modave Lectures on Classical Integrability in 2d Field Theories,” PoS Modave2021(2022) 002,arXiv:2112.14628 [hep-th]

    arXiv 2022

  5. [5]

    Hamiltonian structures for systems of hyperbolic conservation laws,

    P. J. Olver and Y. Nutku, “Hamiltonian structures for systems of hyperbolic conservation laws,” Journal of Mathematical Physics29(1988) no. 7, 1610–1619

    work page 1988

  6. [6]

    Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov-Whitman averaging method,

    B. A. Dubrovin and S. P. Novikov, “Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov-Whitman averaging method,” World Scientific Series in 20th Century Physics11382–386

    work page

  7. [7]

    Integrability properties of Motzkin polynomials

    I. Gahramanov and E. T. Musaev, “Integrability properties of Motzkin polynomials,” J. Math. Phys.61(2020) no. 3, 033509,arXiv:1706.00197 [hep-th]

    work page Pith review arXiv 2020

  8. [8]

    Liouville integrable binomial Hamiltonian system

    M. Mullahasanoglu, “Liouville integrable binomial Hamiltonian system,” J. Phys. Conf. Ser. 2667(2023) no. 1, 012041,arXiv:2304.04500 [nlin.SI]

    work page Pith review arXiv 2023

Show all 51 references
  1. [9]

    An exact solution for a derivative nonlinear Schr¨ odinger equation,

    D. J. Kaup and A. C. Newell, “An exact solution for a derivative nonlinear Schr¨ odinger equation,” Journal of Mathematical Physics19(1978) no. 4, 798–801

    1978

  2. [10]

    Encyclopedia of integrable systems,

    A. Shabat, V. Adler, V. Marikhin, and V. Sokolov, “Encyclopedia of integrable systems,”

  3. [11]

    Nonlinear differential difference equations as Backlund transformations,

    D. Levi, “Nonlinear differential difference equations as Backlund transformations,” Journal of Physics A: Mathematical and General14(1981) no. 5, 1083

    1981

  4. [12]

    The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems,

    A. B. Mikhailov, A. V. Shabat and R. I. Yamilov, “The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems,” Russian Math. Surveys42(1987) no. 4, 1–63

    1987

  5. [13]

    Gauge transformations of constrained KP flows: New integrable hierarchies,

    A. Kundu, W. Strampp, and W. Oevel, “Gauge transformations of constrained KP flows: New integrable hierarchies,” Journal of Mathematical Physics36(1995) no. 6, 2972–2984

    1995

  6. [14]

    A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations,

    E. Fan, “A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations,” Journal of Mathematical Physics42(2001) no. 9, 4327–4344. – 15 –

    2001

  7. [15]

    A systematic literature review of Burgers’ equation with recent advances,

    M. P. Bonkile, A. Awasthi, C. Lakshmi, V. Mukundan, and V. S. Aswin, “A systematic literature review of Burgers’ equation with recent advances,” Pramana90(2018) no. 6, 69

    2018

  8. [16]

    Progresses on Some Open Problems Related to Infinitely Many Symmetries,

    S. Lou, “Progresses on Some Open Problems Related to Infinitely Many Symmetries,” Mathematics12(2024) no. 20, 3224,arXiv:2406.00208 [nlin.SI]

    2024

  9. [17]

    Integrability of Limit Shapes of the Six Vertex Model,

    N. Reshetikhin and A. Sridhar, “Integrability of Limit Shapes of the Six Vertex Model,” Commun. Math. Phys.356(2017) no. 2, 535–565,arXiv:1510.01053 [math-ph]

    2017 arXiv

  10. [18]

    Covariant canonical formulations of classical field theories,

    F. Gieres, “Covariant canonical formulations of classical field theories,” SciPost Phys. Lect. Notes77(2023) 1,arXiv:2109.07330 [hep-th]

    2023 arXiv

  11. [19]

    Hints on integrability in the Wilsonian/holographic renormalization group,

    E. T. Akhmedov, I. B. Gahramanov, and E. T. Musaev, “Hints on integrability in the Wilsonian/holographic renormalization group,” JETP Lett.93(2011) 545–550

    2011

  12. [20]

    Exact statement for Wilsonian and holographic renormalization group,

    E. T. Akhmedov and E. T. Musaev, “Exact statement for Wilsonian and holographic renormalization group,” Phys. Rev. D81(2010) 085010,arXiv:1001.4067 [hep-th]

    2010 arXiv

  13. [21]

    Renormalization and Effective Lagrangians,

    J. Polchinski, “Renormalization and Effective Lagrangians,” Nucl. Phys. B231(1984) 269–295

    1984

  14. [22]

    The Wilson-Polchinski exact renormalization group equation,

    C. Bervillier, “The Wilson-Polchinski exact renormalization group equation,” Phys. Lett. A 332(2004) 93–100,arXiv:hep-th/0405025

    2004 arXiv

  15. [23]

    Multi-Hamiltonian structure of the Born–Infeld equation,

    M. Arik, F. Neyzi, Y. Nutku, P. J. Olver, and J. M. Verosky, “Multi-Hamiltonian structure of the Born–Infeld equation,” Journal of Mathematical Physics30(1989) no. 6, 1338–1344

    1989

  16. [24]

    Perfect fluid theory and its extensions,

    R. Jackiw, V. P. Nair, S. Y. Pi, and A. P. Polychronakos, “Perfect fluid theory and its extensions,” J. Phys. A37(2004) R327–R432,arXiv:hep-ph/0407101

    2004 arXiv

  17. [25]

    The On-Line Encyclopedia of Integer Sequences,

    N. J. A. Sloane, “The On-Line Encyclopedia of Integer Sequences,”

  18. [26]

    “Motzkin paths, Motzkin polynomials and recurrence relations,

    R. Oste and J. V. der Jeugt, ““Motzkin paths, Motzkin polynomials and recurrence relations,” Electronic journal of combinatorics22(2015) no. 2,

    2015

  19. [27]

    Asymptotic integrability of nonlinear wave equations,

    A. M. Kamchatnov, “Asymptotic integrability of nonlinear wave equations,” Chaos: An Interdisciplinary Journal of Nonlinear Science34(2024) no. 11, 113117

    2024

  20. [28]

    Quasiclassical integrability condition in AKNS scheme,

    A. Kamchatnov and D. Shaykin, “Quasiclassical integrability condition in AKNS scheme,” Physica D: Nonlinear Phenomena460(2024) 134085

    2024

  21. [29]

    Lagrangian Approach to Dispersionless KdV Hierarchy,

    A. Choudhuri, B. Talukdar, and U. Das, “Lagrangian Approach to Dispersionless KdV Hierarchy,” Symmetry, Integrability and Geometry: Methods and Applications (2007)

    2007

  22. [30]

    Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method,

    H. H. Chen, Y. C. Lee, and C. S. Liu, “Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method,” Physica Scripta20(1979) no. 3-4, 490

    1979

  23. [31]

    Integrable evolution equations on associative algebras,

    P. J. Olver and V. V. Sokolov, “Integrable evolution equations on associative algebras,” Communications in Mathematical Physics193(1998) no. 2, 245–268

    1998

  24. [32]

    The quadratic bundle of general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures,

    V. S. Gerdjikov and M. I. Ivanov, “The quadratic bundle of general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures,” Bulg. J. Phys.10(1983) 130–143

    1983

  25. [33]

    Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model,

    N. A. Kudryashov, S. F. Lavrova, and D. R. Nifontov, “Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model,” Mathematics 11(2023) no. 23,

    2023

  26. [34]

    Solutions to the wave equation for commuting – 16 – flows of dispersionless PDEs,

    N. Manganaro, A. Rizzo, and P. Vergallo, “Solutions to the wave equation for commuting – 16 – flows of dispersionless PDEs,” International Journal of Non-Linear Mechanics159(2024) 104611

    2024

  27. [35]

    Solitary Waves of Burgers Hierarchy Equations,

    N. A. Kudryashov, “Solitary Waves of Burgers Hierarchy Equations,” Computational Mathematics and Mathematical Physics65(2025) no. 5, 995–1003

    2025

  28. [36]

    The Diophantine equationx 2 −Dy 2 =N,D >0,

    K. R. Matthews, “The Diophantine equationx 2 −Dy 2 =N,D >0,” Expositiones Mathematicae18(2000) no. 4, 323–331

    2000

  29. [37]

    Solving the Pell equation,

    H. W. Lenstra, Jr., “Solving the Pell equation,” Notices of the American Mathematical Society49(2002) no. 2, 182–192

    2002

  30. [38]

    Deformations of dispersionless Lax systems,

    W. Kry´ nski, “Deformations of dispersionless Lax systems,” Classical and Quantum Gravity 40(2023) no. 23, 235013

    2023

  31. [39]

    Integrable equations in 2 + 1 dimensions: deformations of dispersionless limits,

    E. V. Ferapontov, A. Moro, and V. S. Novikov, “Integrable equations in 2 + 1 dimensions: deformations of dispersionless limits,” Journal of Physics A: Mathematical and Theoretical 42(2009) no. 34, 345205

    2009

  32. [40]

    Nonlinear Schr¨ odinger equations of general form and their exact solutions,

    N. A. Kudryashov and A. D. Polyanin, “Nonlinear Schr¨ odinger equations of general form and their exact solutions,” Applied Mathematics Letters170(2025) 109622

    2025

  33. [41]

    Integration of nonlinear system of four waves with two velocities in (2 + 1) dimensions by the inverse scattering transform method,

    M. I. Ismailov, “Integration of nonlinear system of four waves with two velocities in (2 + 1) dimensions by the inverse scattering transform method,” Journal of Mathematical Physics52 (2011) no. 3, 033504

    2011

  34. [42]

    Lagrangian multiforms and dispersionless integrable systems,

    E. V. Ferapontov and M. Vermeeren, “Lagrangian multiforms and dispersionless integrable systems,” Letters in Mathematical Physics115(2025) no. 6, 125

    2025

  35. [43]

    Interpolating Dispersionless Integrable System,

    M. Dunajski, “Interpolating Dispersionless Integrable System,” J. Phys. A41(2008) 315202, arXiv:0804.1234 [nlin.SI]

    2008 arXiv

  36. [44]

    Integrability via geometry: dispersionless differential equations in three and four dimensions,

    D. M. J. Calderbank and B. Kruglikov, “Integrability via geometry: dispersionless differential equations in three and four dimensions,” Commun. Math. Phys.382(2021) no. 3, 1811–1841, arXiv:1612.02753 [math.AP]

    2021 arXiv

  37. [45]

    Integrability Ex Machina,

    S. Krippendorf, D. L¨ ust, and M. Syvaeri, “Integrability Ex Machina,” Fortschritte der Physik 69(2021) no. 7, 2100057,arXiv:2103.07475 [nlin.SI]

    2021 arXiv

  38. [46]

    Classical Integrability in the Presence of a Cosmological Constant: Analytic and Machine Learning Results,

    G. Lopes Cardoso, D. Mayorga Pena, and S. Nampuri, “Classical Integrability in the Presence of a Cosmological Constant: Analytic and Machine Learning Results,” Fortsch. Phys.73(2025) no. 4, 2400267,arXiv:2404.18247 [hep-th]

    2025 arXiv

  39. [47]

    Machine-Learning Search for Lax Connections,

    O. Fukushima, T. Shigemura, R. Suda, N. Tanahashi, and K. Yoshida, “Machine-Learning Search for Lax Connections,”arXiv:2608.05146 [hep-th]

    arXiv

  40. [48]

    Direct poisson neural networks: learning non-symplectic mechanical systems,

    M. ˇS ´ ıpka, M. Pavelka, O. Esen, and M. Grmela, “Direct poisson neural networks: learning non-symplectic mechanical systems,” Journal of Physics A: Mathematical and Theoretical56 (2023) no. 49, 495201

    2023

  41. [49]

    Gauge Theory And Integrability, III,

    K. Costello and M. Yamazaki, “Gauge Theory And Integrability, III,”arXiv:1908.02289 [hep-th]

    1908 arXiv

  42. [50]

    Dualities and discretizations of integrable quantum field theories from 4d Chern-Simons theory,

    M. Ashwinkumar, J.-i. Sakamoto, and M. Yamazaki, “Dualities and discretizations of integrable quantum field theories from 4d Chern-Simons theory,” Adv. Theor. Math. Phys. 29(2025) no. 6, 1509–1694,arXiv:2309.14412 [hep-th]

    2025

  43. [51]

    Gauge Theory and Integrability: An Overview,

    M. Yamazaki, “Gauge Theory and Integrability: An Overview,”arXiv:2509.07628 [hep-th]. – 17 –

    arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.