Pith. sign in

REVIEW 2 major objections 5 minor 3 cited by

From the self-dual Yang-Mills equation to the Fokas-Lenells equation

T0 review · 2 major / 5 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 Self-dual Yang-Mills reduces to the Fokas-Lenells equation via two Cauchy-matrix schemes, yielding explicit N-soliton solutions.

desk verdict Solid SDYM-to-FL reduction with checkable Cauchy-matrix solitons, but the claimed equivalence of the two solution schemes is proved only for the alternative AKNS pair, not the primary one used later. read the letter →

arxiv 2411.10807 v1 pith:4GVXAZ2E submitted 2024-11-16 nlin.SI math-phmath.MP

Shangshuai Li , Shuzhi Liu , Da-jun Zhang This is my paper · ORCID
classification nlin.SImath-phmath.MP MSC 37K1035Q5137K35
keywords self-dualYang-MillsequationFokas-LenellspKN(-1)systemCauchymatrixapproachWardconjectureN-solitonsolutionsKaup-Newellhierarchyconjugatereduction
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 seeks to show that the four-dimensional self-dual Yang-Mills (SDYM) equation, in its K-matrix form, reduces under a specific SL(2) gauge choice and a $\sigma_3$-twist in the $w$-direction to the unreduced Fokas-Lenells system, the pKN(-1) system $u_{xt}-u-2iuvu_x=0$, $v_{xt}-v+2ivuv_x=0$. The reduction is then realized inside two previously built Cauchy matrix solution schemes, one from the matrix KP hierarchy and one from the AKNS hierarchy, giving explicit determinant formulas for solutions. A conjugate reduction $v=u^*$ turns these into N-soliton solutions of the Fokas-Lenells equation itself. The two schemes' solutions are shown equivalent under the reflection $(x,t)\mapsto(-x,-t)$. A sympathetic reader would care because the paper supplies a new, explicit instance of the conjecture that integrable equations are reductions of SDYM, and it points toward a Cauchy matrix description of the Kaup-Newell hierarchy.

What carries the argument

The carrying object is the general (K-matrix/Miura) formulation of SDYM, equation (2.8), together with the reduction ansatz (2.10): gauge group SL(2), $\tilde w=w$, and $J,K$ twisted by $\sigma_3$ in $w$. The reduction works because the twist turns $w$-derivatives into commutators $[\,\cdot\,,\sigma_3]$, so the four-dimensional equation collapses to two-dimensional relations among the entries of $J$ and $K$. To generate solutions, the paper uses the Cauchy matrix master functions $S^{(i,j)}$ defined from Sylvester equations (3.1) and (3.4): these matrices already satisfy the SDYM equation and the commutator identities needed for the ansatz, and they encode the solutions as ratios of determinants. Conjugate reduction is imposed by constraints on the spectral matrices and wave factors, namely $L=-K^\dagger$ with $s_1^T=-ir_1^\dagger K^\dagger$, $s_2^T=r_2^\dagger$ in the KP scheme and $K_2=-K_1^\dagger$, $s_2=r_1^*$, $r_2=iK_1^* s_1^*$ in the AKNS scheme, which force $v=u^*$.

What would settle it

Compute the Fokas-Lenells residual for the one-soliton formulas (4.6) and (4.11) at a generic set of parameters $k_1,\lambda_1,\lambda_2$ with high-precision arithmetic; any nonzero residual at a regular point would falsify the claim. A second check is whether the reflection equivalence (3.54) survives when $K$ is a Jordan block (multiple-pole case), since the proof in Section 3.2.3 is written for diagonal $K$.

Watch

Extended reading notes

Core claim

From the general SDYM equation $\partial_{\tilde z}K = -(\partial_w J)J^{-1}$, $\partial_{\tilde w}K = -(\partial_z J)J^{-1}$ with gauge group SL(2), real coordinates with $\tilde w=w$, and $J=e^{-\sigma_3 w}J' e^{\sigma_3 w}$, $K=e^{-\sigma_3 w}K' e^{\sigma_3 w}$, the paper proves that $(u,v)=(K_{21},\, iJ_{12}/J_{22})$ satisfies the pKN(-1) system. The same reduction is realizable in both the KP-type and AKNS-type Sylvester-equation schemes, where the master functions $S^{(i,j)}$ supply $J$ and $K$; the resulting pKN(-1) solutions take explicit rational-determinant form. Under the conjugate constraints (4.2) and (4.8), $v=u^*$ holds and the formulas become N-soliton solutions of the Fokas-Lenells equation $u_{xt}-u-2i|u|^2u_x=0$. The paper also shows that the two solution families are related by the reflection $(u^{[KP]}(x,t),v^{[KP]}(x,t))=(v^{[AKNS]}(-x,-t),u^{[AKNS]}(-x,-t))$, and that the displayed solutions coincide with the bright soliton solutions known from bilinear theory.

Load-bearing premise

The paper assumes, without re-deriving, the previously established Cauchy matrix formulations of the SDYM equation: that the master functions $S^{(i,j)}$ satisfy the general SDYM equation and give $\det V=1$; if that foundation fails for some spectral choices, the explicit pKN(-1) and FL solutions are not guaranteed.

Editorial extensions

If this is right

  • Explicit N-soliton solutions of the Fokas-Lenells equation follow from either scheme, and Appendix B shows they match the known bright-soliton solutions from the bilinear method.
  • The pKN(-1) solutions from the KP-type and AKNS-type schemes are interchangeable under the reflection $(x,t)\mapsto(-x,-t)$, so results proved in one scheme transfer to the other.
  • The intermediate AKNS(-1) system (2.22) appears naturally in the reduction and is connected to pKN(-1) by the Miura relation (5.1), giving a bridge between the two hierarchies.
  • The same Cauchy matrix data, with Jordan-block spectral matrices, produce multiple-pole solutions of pKN(-1) and of the Fokas-Lenells equation, as constructed in Appendix A.
  • A nonlocal reduction gives the nonlocal Fokas-Lenells equation (4.15) and its solutions, widening the applicability of the scheme.

Reading between the lines

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

  • The same reduction recipe with any of the projections $P_1,P_2,P_3$ replacing $\sigma_3$ suggests a family of related reductions; testing whether other members of the Kaup-Newell negative hierarchy descend from SDYM by the same mechanism would be a natural next step.
  • The reflection equivalence between the two Cauchy schemes hints that the conjugate and nonlocal reductions could be composed, producing solutions with prescribed parity under $(x,t)\mapsto(-x,-t)$ without extra computation.
  • Because the multiple-pole construction in Appendix A is carried out only in the KP-type scheme, extending the Jordan-block argument to the AKNS-type scheme would likely yield second families of rational solutions, a straightforward but unstated corollary.
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

2 major / 5 minor

Summary. The paper derives the unreduced Fokas-Lenells system, the pKN(-1) system (1.3), as a dimensional reduction of the general self-dual Yang-Mills equation in K-matrix form (2.8), under the reduction constraints (2.10). Theorem 1 gives the reduction explicitly. Section 3 realizes the reduction in two Cauchy matrix schemes, the KP-type and the AKNS-type, and presents explicit solution formulas for pKN(-1) in (3.35) and (3.45)/(3.47). Proposition 1 claims that these two solution families are equivalent under the reflection (x,t) -> (-x,-t). Section 4 applies conjugate reductions v = u* under constraints (4.1) and (4.7), yielding N-soliton formulas for the Fokas-Lenells equation in Theorems 3 and 5. Appendix A constructs multiple-pole solutions, and Appendix B compares the FL formulas with Matsuno's bilinear solutions.

Significance. If the technical gaps noted below are repaired, this is a useful contribution: it adds a direct example to Ward's conjecture connecting SDYM to the Fokas-Lenells equation via pKN(-1), and it gives explicit Cauchy-matrix solutions for the FL equation and the Kaup-Newell hierarchy. The main reduction in Theorem 1 is clean and self-contained, and the displayed solution formulas are concrete and reproducible. The comparison with Matsuno's bilinear solutions in Appendix B is strong supporting evidence that the final formulas are correct. The main obstacles are the unproved equivalence in Proposition 1 and an incorrect intermediate identity in the proof of Theorem 2; both are local and fixable, but they affect claims that are central to the paper.

major comments (2)
  1. [§3.2.3, Proposition 1] Proposition 1 is not proved for the AKNS-type solution that is actually used in Section 4. The proof begins with the alternative solution (3.47) of Remark 7 and shows, for that pair, that (u_AKNS(-x,-t), v_AKNS(-x,-t)) equals (v_KP(x,t), u_KP(x,t)). The primary AKNS pair (3.40)/(3.45), with u = u_3 and v = i v_2/v_4, is the one used in Theorem 5 via (4.11), but no argument connects this primary pair to (3.47) or to the KP pair under the reflection. As the text itself notes, it recovers the KP-type solution from the AKNS-type solution given in (3.47). Therefore the assertion that solutions derived from the AKNS-type scheme in Section 3.2.2 are equivalent to the KP-type solutions is broader than what is proved. Please extend the proof to the primary pair or restrict the proposition and re-examine the statements in Sections 4 and 5 that depend on it.
  2. [§4.1, proof of Theorem 2] The proof contains a false intermediate statement: after applying (4.2) to (3.19d), the text says 'one has M = M†'. For the matrix M defined by (4.4b) this is not true; already in the N=1 case, M - M† = -i |ρ1(k1)|^2 != 0. The subsequent identity K M + M† K† = r2 r2† is nevertheless true, and it is sufficient for the rest of the calculation, since the step from the second line to the third line uses this identity rather than M = M†. Please replace the incorrect claim with the correct identity and adjust the derivation accordingly.
minor comments (5)
  1. [§3.1, equation (3.6)] The second Sylvester equation is displayed as K2M2 - M2K2 = r2s1^T; the left-hand side should presumably be K2M2 - M2K1, as used later in (4.9).
  2. [§3.2.2, equation (3.47a)] The displayed formula for u contains a typographical artifact reading 'u = i −s(−1, 0)_3 ...'; please clean up the formula and the surrounding notation.
  3. [§3.2.2] The phrase 'and the Remark 6 holds too' is ambiguous; please state explicitly that the plane-wave factors (3.26) are used in the AKNS-type construction.
  4. [§3.2.1 and §3.2.2] The same symbols u and v are used for the pKN(-1) fields, for the matrices U and V, and for the entries of J and K; please introduce a clearer distinction or explicitly say 'entry' when applying (2.18) to (3.24).
  5. [Appendix B] The verification gives u = -u*[KP] (and similarly for u[AKNS]) rather than equality; this is acceptable as an external check, but the sign and conjugation conventions should be stated explicitly so the reader does not interpret the comparison as exact equality.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SDYM-to-pKN(-1) reduction is a direct derivation from stated constraints, and the Cauchy-matrix and soliton formulas are supported by the external Matsuno comparison.

full rationale

The central claim, Theorem 1, derives the pKN(-1) system (1.3) from the general SDYM equation (2.8) under the explicitly stated constraints (2.10), with an algebraic verification in (2.17)-(2.21). This is a parameter-free reduction, not a fit or a definitional identity. The realization of the constraints inside the KP- and AKNS-type Cauchy matrix schemes (Sec. 3.1) rests on stated Sylvester equations and dispersion relations; the invoked results from the authors' prior papers [26,27] provide the master-function identities and the |J|=1 normalization. These citations are load-bearing for the solution construction, but they are not circular: they are stated, checkable Sylvester-equation computations, and the resulting FL soliton formulas are explicitly compared with Matsuno's bilinear solutions in Appendix B, an external benchmark independent of the present paper's fitted values. The conjugate reductions in Theorems 2 and 4 are direct algebraic consequences of the imposed constraints (4.1) and (4.7), with proofs given in the text. One rigor concern, unrelated to circularity, is that the equivalence proof in Sec. 3.2.3 starts from the alternative AKNS solution (3.47) rather than the primary construction (3.45), and the text says 'we recover the KP-type solution ... from the AKNS-type solution ... given in (3.47)'; Proposition 1 as stated is therefore not fully proven for the main AKNS formula. This is an omitted-proof or mislabeling issue, not a reduction of the result to its inputs. Accordingly, no circular step can be exhibited, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central derivation is self-contained, but it relies on the Cauchy matrix machinery from [26,27] and on standard identities for Cauchy matrices. There are no parameters fitted to data and no invented entities; the arbitrary spectral parameters k_i, l_i and phase factors are part of the solution family, not free parameters of the derivation.

assumptions (3)
  • domain assumption The KP-type and AKNS-type Cauchy matrix master functions satisfy the general SDYM equation (2.8) and produce det(V)=1 (Theorem 2 in [27]).
    Used in Section 3.1 to realize the reduction; results are cited from the authors' previous paper [27] rather than re-derived here.
  • standard math The identity G^{-1} = X G^T Y with X, Y diagonal (3.48)-(3.49), and the Lagrange property G X e_N = e_N (3.50), hold for the Cauchy matrix G.
    Invoked in Section 3.2.3 to prove equivalence of KP-type and AKNS-type solutions; sourced from Lemma 2.4 in [40].
  • domain assumption The bilinear determinant formulas of Matsuno [20] give the same FL solutions as the Cauchy matrix construction (Appendix B).
    Used as an external consistency check; the paper states the correspondence but does not prove Matsuno's results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of From the self-dual Yang-Mills equation to the Fokas-Lenells equation." pith.science (2026). https://pith.science/paper/4GVXAZ2E

@misc{pith2026241110807,
  author       = {Pith},
  title        = {Pith review of: From the self-dual Yang-Mills equation to the Fokas-Lenells equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GVXAZ2E}},
  note         = {Machine review of arXiv:2411.10807}
}
read the original abstract

A reduction from the self-dual Yang-Mills (SDYM) equation to the unreduced Fokas-Lenells (FL) system is described in this paper. It has been known that the SDYM equation can be formulated from the Cauchy matrix schemes of the matrix Kadomtsev-Petviashvili (KP) hierarchy and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. We show that the reduction can be realized in these two Cauchy matrix schemes, respectively. Each scheme allows us to construct solutions for the unreduced FL system. We prove that these solutions obtained from different schemes are equivalent under certain reflection transformation of coordinates. Using conjugate reduction we obtain solutions of the FL equation. The paper adds an important example to Ward's conjecture on the reductions of the SDYM equation. It also indicates the Cauchy matrix structures of the Kaup-Newell hierarchy.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Miura transformation in bidifferential calculus and a vectorial Darboux transformation for the Fokas-Lenells equation

    nlin.SI 2025-04 conditional novelty 7.0 of 10

    A vectorial binary Darboux transformation for the coupled Fokas-Lenells equations is derived from bidifferential calculus and reduced to the scalar Fokas-Lenells equation, generating breathers, dark/bright solitons, a...

  2. Asymptotic Equivalence Between Quasi-Grammian and Quasi-Wronskian $N$-Soliton Solutions of the Anti-Self-Dual Yang-Mills Equation

    nlin.SI 2026-07 conditional novelty 6.0 of 10

    Quasi-Grammian and quasi-Wronskian N-soliton solutions of the ASDYM/Yang equation are asymptotically equivalent up to a constant matrix factor, with explicit N-soliton phase shifts.

  3. Soliton Resonances in Four Dimensional Wess-Zumino-Witten Model

    hep-th 2025-01 conditional novelty 6.0 of 10

    Double-pole and V-shape resonance soliton solutions are constructed for the U(2) Yang equation on ultrahyperbolic space, with action densities that match known one- and two-soliton results and suggest new building blo...

Reference graph

Works this paper leans on

45 extracted references · 43 canonical work pages · cited by 3 Pith papers

  1. [1]

    Yang, Condition of self-duality for SU(2) gauge fields on Euclid ean four-dimensional space, Phys

    C.N. Yang, Condition of self-duality for SU(2) gauge fields on Euclid ean four-dimensional space, Phys. Rev. Lett., 38 (1977) 1377-1379

    work page 1977

  2. [2]

    Donaldson, Anti self-dual Yang-Mills connections over comp lex algebraic surfaces and stable vector bundles, Proc

    S.K. Donaldson, Anti self-dual Yang-Mills connections over comp lex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc., 50 (1985) 1-26

    work page 1985

  3. [3]

    Mason, N.M.J

    L.J. Mason, N.M.J. Woodhouse, Integrability, Self-Duality, and Tw istor Theory, Oxford University Press, Oxford, New York, 1996

    work page 1996

  4. [4]

    Abiowitz, S

    M.J. Abiowitz, S. Chakravarty, R.G. Halburd, Integrable system s and reductions of the self-dual Yang-Mills equations, J. Math. Phys., 44 (2003) 3147-3173

    work page 2003

  5. [5]

    Mason, N.M.J

    L.J. Mason, N.M.J. Woodhouse, Self-duality and the Painlev´ e tran scendents, Nonlinearity, 6 (1993) 569-581

    work page 1993

  6. [6]

    Manakov, V.E

    S.V. Manakov, V.E. Zakharov, Three-dimensional model of relat ivistic-invariant field theory, inte- grable by the inverse scattering transform, Lett. Math. Phys., 5 (1981) 247-253

    work page 1981

  7. [7]

    Ward, Integrable and solvable systems, and relations among them, Philos

    R.S. Ward, Integrable and solvable systems, and relations among them, Philos. Trans. R. Soc. Lond. A, 315 (1985) 451-457

    work page 1985

  8. [8]

    Fokas, On a class of physically important integrable equations , Physica D, 87 (1995) 145-150

    A.S. Fokas, On a class of physically important integrable equations , Physica D, 87 (1995) 145-150

    work page 1995

Show all 45 references
  1. [9]

    Lenells, A.S

    J. Lenells, A.S. Fokas, On a novel integrable generalization of the nonlinear Schr¨ odinger equation, Nonlinearity, 22 (2009) 11-27

    2009

  2. [10]

    Lenells, Exactly solvable model for nonlinear pulse propagatio n in optical fibers, Stud

    J. Lenells, Exactly solvable model for nonlinear pulse propagatio n in optical fibers, Stud. Appl. Math., 123 (2009) 215-232. 21

    2009

  3. [11]

    Lenells, Dressing for a novel integrable generalization of the nonlinear Schr¨ odinger equation, J

    J. Lenells, Dressing for a novel integrable generalization of the nonlinear Schr¨ odinger equation, J. Nonl. Sci., 20 (2010) 709-722

    2010

  4. [12]

    Mikhailov, Integrability of the two-dimensional Thirring mode l, JETP Lett., 23 (1976) 320-323

    A.V. Mikhailov, Integrability of the two-dimensional Thirring mode l, JETP Lett., 23 (1976) 320-323

    1976

  5. [13]

    Gerdjikov, M.I

    V.S. Gerdjikov, M.I. Ivanov, P.P. Kulish, Quadratic bundle and no nlinear equations, Theor. Math. Phys., 44 (1980) 784-795

    1980

  6. [14]

    Gerdjikov, M.I

    V.S. Gerdjikov, M.I. Ivanov, The quadratic pencil of general t ype and the nonlinear evolution. Hierarchies of Hamiltonian structures, JINR preprint E2-82-595, Dubna, USSR (1982) (17pp)

    1982

  7. [15]

    Kaup, A.C

    D.J. Kaup, A.C. Newell, On the Coleman correspondence and the s olution of the massive Thirring model, Lett. AL Nuovo Cimento, 20 (1977) 325-331

    1977

  8. [16]

    S.Z. Liu, J. Wang, D.J. Zhang, The Fokas-Lenells equations: Biline ar approach, Stud. Appl. Math., 148 (2022) 651-688

    2022

  9. [17]

    Zhao, E.G

    Y. Zhao, E.G. Fan, Inverse scattering transformation for th e Fokas-Lenells equation with nonzero boundary conditions, J. Nonl. Math. Phys., 28 (2021) 38-52

    2021

  10. [18]

    L.P. Ai, J. Xu, On a Riemann-Hilbert problem for the Fokas-Lenells equation, Appl. Math. Lett., 87 (2019) 57-63

    2019

  11. [19]

    Zhang, S.F

    X.F. Zhang, S.F. Tian, Riemann-Hilbert problem for the Fokas-Le nells equation in the presence of high-order discrete spectrum with non-vanishing boundary condit ions, J. Math. Phys., 64 (2023) 051503 (18pp)

    2023

  12. [20]

    Matsuno, A direct method of solution for the Fokas-Lenells d erivative nonlinear Schr¨ odinger equation: I

    Y. Matsuno, A direct method of solution for the Fokas-Lenells d erivative nonlinear Schr¨ odinger equation: I. Bright soliton solutions, J. Phys. A: Math. Theor., 45 ( 2012) 235202 (19pp)

    2012

  13. [21]

    Matsuno, A direct method of solution for the Fokas-Lenells d erivative nonlinear Schr¨ odinger equation: II

    Y. Matsuno, A direct method of solution for the Fokas-Lenells d erivative nonlinear Schr¨ odinger equation: II. Dark soliton solutions, J. Phys. A: Math. Theor., 45 ( 2012) 475202 (31pp)

    2012

  14. [22]

    Liu, C.C

    F.F. Liu, C.C. Zhou, X.L¨ u, H.T. Xu, Dynamic behaviors of optical s olitons for Fokas-Lenells equa- tion in optical fiber, Optik, 224 (2020) 165237 (9pp)

    2020

  15. [23]

    J.S. He, S.W. Xu, K. Porsezian, Rogue waves of the Fokas-Lene lls equation, J. Phys. Soc. Jpn. 81 (2012) 124007 (4pp)

    2012

  16. [24]

    Wang, Z.J

    Y. Wang, Z.J. Xiong, L. Ling, Fokas-Lenells equation: Three typ es of Darboux transformation and multi-soliton solutions, Appl. Math. Lett., 107 (2020) 106441 (8pp)

    2020

  17. [25]

    R.S. Ye, Y. Zhang, A vectorial Darboux transformation for th e Fokas-Lenells system, Chaos Solitons Fractals, 169 (2023) 113233 (7pp)

    2023

  18. [26]

    S.S. Li, C.Z. Qu, X.X. Yi, D.J. Zhang, Cauchy matrix approach to th e SU(2) self-dual Yang-Mills equation, Stud. Appl. Math., 148 (2022) 1703-1721

    2022

  19. [27]

    S.S. Li, C.Z. Qu, D.J. Zhang, Solutions to the SU(N) self-dual Yan g-Mills equation, Physica D, 453 (2023) 133828 (17pp)

    2023

  20. [28]

    Nijhoff, J

    F.W. Nijhoff, J. Atkinson, J. Hietarinta, Soliton solutions for ABS lattice equations: I. Cauchy matrix approach, J. Phys. A: Math. Theor., 42 (2009) 404005, (3 4pp)

    2009

  21. [29]

    Adler, A.I

    V.E. Adler, A.I. Bobenko, Yu.B Suris, Classification of integrable e quations on quad-graphs, the consistency approach, Commun. Math. Phys., 233 (2002) 513-54 3

    2002

  22. [30]

    Zhang, S.L

    D.J. Zhang, S.L. Zhao, Solutions to ABS lattice equations via gene ralized Cauchy matrix approach, Stud. Appl. Math., 131 (2013) 72-103

    2013

  23. [31]

    D.D. Xu, D.J. Zhang, S.L. Zhao, The sylvester equation and integ rable equations: I. The Korteweg- de Vries system and sine-Gordon equation, J. Nonlinear Math. Phys ., 21 (2014) 382-406

    2014

  24. [32]

    Feng, S.L

    W. Feng, S.L. Zhao, The Sylvester equation and Kadomtsev-Pe tviashvili system, Symmetry, 14 (2022) 542 (17pp). 22

    2022

  25. [33]

    Zhao, The Sylvester equation and integrable equations: Th e Ablowitz-Kaup-Newell-Segur sys- tem, Rep

    S.L. Zhao, The Sylvester equation and integrable equations: Th e Ablowitz-Kaup-Newell-Segur sys- tem, Rep. Math. Phys., 82 (2018) 241-263

    2018

  26. [34]

    Hamanaka, S.C

    M. Hamanaka, S.C. Huang, New soliton solutions of anti-self-dua l Yang-Mills equations, J. High Energ. Phys., 2020 (2020) 101 (18pp)

    2020

  27. [35]

    Huang, On Soliton Solutions of the Anti-Self-Dual Yang-Mills E quations from the Perspective of Integrable Systems (PhD thesis), Nagoya University, 2021, ar Xiv:2112.10702 [hep-th]

    S.C. Huang, On Soliton Solutions of the Anti-Self-Dual Yang-Mills E quations from the Perspective of Integrable Systems (PhD thesis), Nagoya University, 2021, ar Xiv:2112.10702 [hep-th]

    2021 arXiv

  28. [36]

    S.S. Li, D.J. Zhang, Direct linearization of the SU(2) anti-self-du al Yang-Mills equation in various spaces, to appear in J. Geom. Phys., (2024), arXiv: 2403.06055 [nlin .SI]

    2024 arXiv

  29. [37]

    Nimmo, C.R

    J.J.C. Nimmo, C.R. Gilson, Y.Ohta, Applications of Darboux transfo rmations to the self-dual Yang- Mills equations, Theor. Math. Phys., 122 (2000) 239-246

    2000

  30. [38]

    Pohlmeyer, On the Lagrangian theory of anti-self-dual field s in four-dimensional Euclidean space, Commun

    K. Pohlmeyer, On the Lagrangian theory of anti-self-dual field s in four-dimensional Euclidean space, Commun. Math. Phys., 72 (1980) 37-47

    1980

  31. [39]

    Zhang, J

    D.J. Zhang, J. Ji, S.L. Zhao, Soliton scattering with amplitude cha nges of a negative order AKNS equation, Physica D, 238 (2009) 2361-2367

    2009

  32. [40]

    Lynch, Cauchy pairs and Cauchy matrices, Linear Algebra A ppl., 471 (2015) 320-345

    A.G. Lynch, Cauchy pairs and Cauchy matrices, Linear Algebra A ppl., 471 (2015) 320-345

    2015

  33. [41]

    Atiyah, R.S

    M.F. Atiyah, R.S. Ward, Instantons and algebraic geometry, Co mmun. Math. Phys., 55 (1977) 117-124

    1977

  34. [42]

    Atiyah, N.J

    M.F. Atiyah, N.J. Hitchin, V.G. Drinfield, Y.I. Manin, Constriction of instantons, Phys. Lett. A, 65 (1978) 185-187

    1978

  35. [43]

    A.A. Cho, J. Wang, D.J. Zhang, Discretization of the modified Kor teweg-de Vries-sine Gordon equation, Theore. Math. Phys., 217 (2023) 1700-1716

    2023

  36. [44]

    Capel, G.R.W

    F.W.Nijhoff, H.W. Capel, G.R.W. Quispel, Integrable lattice version of the massive Thirring model and its linearization, Phys. Lett. A, 98 (1983) 83-86

    1983

  37. [45]

    Hirota, A new form of B¨ acklund transformations and its relation to the inverse scattering problem, Prog

    R. Hirota, A new form of B¨ acklund transformations and its relation to the inverse scattering problem, Prog. Theo. Phys., 52 (1974), 1498-512. 23

    1974

Pith tools

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