Explicit higher order rational rogue waves of the nonlinear Schr\"odinger equation

ENS Paris-Saclay; Robert Conte; The University of Hong Kong

REVIEW 3 major objections 3 minor 23 references

Explicit higher order rational rogue waves of the nonlinear Schr\"odinger equation

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read This paper claims that the N-th rational rogue wave of the nonlinear Schrödinger equation is generated by a three-term recurrence, yielding the seventh wave explicitly with all six complex parameters.

desk verdict Useful computational reduction with an explicit N=7 rogue wave, but the load-bearing three-term recurrence is asserted without proof and verified only through N=4, so the correct verdict is conditional: send to referees, require a proof or an independent residual check. read the letter →

arxiv 2607.23151 v1 pith:M34BL27U submitted 2026-07-25 math-ph math.MPnlin.SI

Robert Conte , ENS Paris-Saclay , The University of Hong Kong This is my paper · ORCID

classification math-phmath.MPnlin.SI MSC 35Q5537K1035C08 PACS 02.30.Ik02.30.Jr02.30.-f

keywords nonlinearSchrödingerequationrationalroguewavesthree-termrecurrenceWronskiansDarbouxtransformationsuperpositionformulaPeregrinewavehigher-orderbreathers

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 the hierarchy of rational rogue-wave solutions of the nonlinear Schrödinger equation can be generated by a three-term recurrence, so that the N-th wave requires only three determinants of order N−1 instead of two determinants of order 2N. The central identity writes the numerator and denominator of the wave as a bilinear product of two affine polynomials plus the square of the previous wave divided by the one before it. If this identity holds for every N, the author obtains, for the first time, explicit compact polynomial expressions for the seventh rogue wave carrying all six arbitrary complex parameters. The practical point is that these short expressions make it feasible to search for new rogue-wave patterns beyond the already observed concentric rings and polygonal configurations.

What carries the argument

The key object is the canonical Wronskian representation: two 2N×2N matrices whose determinants are the numerator and denominator of the wave. After row transpositions the matrices split into four N×N blocks with nonsingular triangular blocks, and the classical Schur determinant formula reduces them to N-th order determinants. The recurrence then rests on an elimination identity: treating the numerator and denominator as bilinear functions of the two complex-conjugate parameters α_{N−1} and β_{N−1}, and eliminating those parameters, the remainder is the square of the previous wave divided by the one before it. The four affine canonical polynomials are defined as scaled determinants of deriva

What would settle it

Symbolically verify identity (19) for N=5 with generic parameters: compute both sides from the Wronskian definitions and check that the difference vanishes and that divisibility by D_3 and N_3 holds. Independently, substitute the N=7 quotient into the NLS equation and check that the residual simplifies to zero.

Watch

Extended reading notes

Core claim

The paper's central claim is that the N-th rational rogue wave of NLS, normalized as u_N = (N_N/D_N)e^{iT/2}, is produced by two independent three-term recurrences: D_N = R_s R_d + D_{N−1}^2/D_{N−2} and N_N = C_s C_d + N_{N−1}^2/N_{N−2}, where the four canonical polynomials are affine in the new complex parameters and only three of them need be computed because two are conjugate. The author presents this as an exact identity obtained by eliminating the two parameters α_{N−1}, β_{N−1}, and verifies it for N=2,3,4 before applying it to construct N=7. The result is a large compression: the seventh wave, with its six arbitrary complex parameters, is given by polynomials rather than by the order-

Load-bearing premise

The elimination identity behind the recurrence is asserted without a general proof and checked only for N=2,3,4; if the quotient D_{N−1}^2/D_{N−2} or N_{N−1}^2/N_{N−2} is not a polynomial for some N≥5, the recurrence fails and the N=7 expressions are not rogue waves.

Editorial extensions

If this is right

For every N for which the recurrence holds, the N-th rogue wave can be computed from three order N−1 determinants, reducing algebra time and storage compared with order-2N determinants.
The explicit seventh-order wave with all six arbitrary complex parameters is now available in data files, so its pattern landscape can be explored without recomputing large determinants.
The recurrence separates numerator and denominator computation into identical three-term recursions, which can be iterated to arbitrary order in a few lines of computer algebra.
Because the canonical polynomials are affine in the newest parameters, studying how patterns change as a single parameter varies becomes tractable.
If the identity is exact, it supplies the general construction that the nonlinear superposition formula was unable to provide.

Reading between the lines

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

Beyond the paper: the recurrence has the shape of a discrete determinant identity, which suggests the N=5 case should be checkable by a Schur-complement argument; doing so would close the only gap in the paper.
A testable extension: the same parameter-elimination idea could be tried on the vector NLS system mentioned in the conclusion, where the reported obstruction is the size of the determinants.
Because the seventh-order expressions are now explicit, one could numerically scan the six complex parameters for patterns that are neither concentric rings nor polygonal configurations, which is the paper's stated motivation.

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.

Desk Editor's Note

read the letter

The genuinely new thing here is the three-term recurrence (19), which lets you build N_N and D_N from three determinants of order N-1 instead of two of order 2N, and the fully parameterized N=7 wave that comes out of it. That is a real step forward for an active subfield: previous explicit formulas stopped at N=4 or needed parameter constraints, and Gaillard's N=6 was too big to publish. The block-reduction setup in Section II is standard and clearly explained, and the supplementary data files give concrete expressions that others can use. Credit where due: the compact polynomial representation, the explicit N=7 output, and the honest acknowledgment that the old determinant formulas become impractical are all good contributions.

The soft spot is exactly where the stress-test note points. Equation (19) is introduced with 'When one eliminates the two parameters ... it turns out' and then checked for N=2,3,4. There is no general proof that the fractions D_{N-1}^2/D_{N-2} and N_{N-1}^2/N_{N-2} are polynomials, and no argument that the identity holds for all N. The N=7 expressions are built on that identity, so if (19) fails at N=5 or N=6, the whole edifice collapses. The paper does not provide a substitution check of the N=7 polynomials into NLS, and the supplementary files, though large, are not accompanied by a machine-readable residual verification. The claim that the nonlinear superposition formula cannot generate the sequence is cited from earlier work and is probably fine, but it is not central to the new result.

That said, this is not a case where the central argument is obviously wrong. The recurrence is an algebraic reorganization of the same Wronskians, not a fit, and the N=2-4 checks are consistent. The identity may well be true for all N; the problem is that the paper asserts it rather than proving or independently verifying it. This is a fixable issue: a proof of the polynomiality and of the recurrence, or a computer-assisted check that the N=7 rational function satisfies NLS, would turn the conditional verdict into acceptance.

A serious referee should engage with this. It is a useful paper for people who need explicit high-order rogue waves and for anyone interested in compact representations of these solutions. I would not cite it until the recurrence is either proved or independently checked, but I would definitely bring it to a reading group to discuss what it would take to close the gap. Send it to peer review with a request for that proof or verification.

Referee Report

3 major / 3 minor

Summary. The paper proposes a three-term recurrence relation for the numerator and denominator polynomials of the N-th rational rogue wave of the NLS equation. The recurrence (19) is stated in Section III, and the author claims it allows computing the N=7 wave with six arbitrary complex parameters using only three (N-1)-order determinants per step. Section II details a block reduction of the Wronskian representation; Sections IV and V discuss other recurrence relations and alternative representations. Explicit polynomials up to N=3 are given in Appendix A, and larger expressions up to N=7 are provided in supplementary Maple files.

Significance. If (19) is valid, this is a substantial computational improvement over existing determinant methods, and the explicit N=7 wave with six arbitrary complex parameters would be a new result. The paper is honest in providing the raw data files, and the N=2-4 checks are consistent. However, the central identity is not proven and the N=7 output is not independently verified; the contribution is therefore conditional.

major comments (3)

[Section III, Eq. (19)] The three-term recurrence is the central computational claim, but it is introduced by "When one eliminates ... it turns out" and no proof is given. It is verified only for N=2,3,4 in Eqs. (20)-(22). The fractions D_{N-1}^2/D_{N-2} and N_{N-1}^2/N_{N-2} are asserted to be polynomials without demonstration. Since the N=5-7 expressions are generated recursively from this identity, the all-N claim is not established. Please provide a general derivation of (19) or an independent verification for N=5,6,7.
[Section VII] The supplementary material contains Maple data for N=5,6,7 but no evidence is presented that the computed N_N and D_N satisfy the NLS equation. A machine-checkable residual certificate (e.g., substituting into iu_T + u_XX + |u|^2 u = 0 and simplifying to zero) should be included for at least N=7, and ideally for N=5,6. Without this, the explicit N=7 wave cannot be fully validated.
[Section III, Eq. (16)] The canonical representation (16) asserts that N_N and D_N are bilinear in the two complex parameters and that the fractions N_r^N/λ_{N-1}, D_r^N/μ_{N-1} reduce to polynomials. This is plausible from the Wronskian structure but is not demonstrated for general N; it is part of the same unproved elimination that underlies (19). A proof of these polynomiality statements is necessary to justify the recurrence.

minor comments (3)

[Section III, Eq. (18) and example (25)] The constant K_N is defined in (17), but the example after (23)-(25) writes det∂β1 M1 = K_1^2 det(Rs1). Since (18) defines Rs_{N-1}=K_N^{-2} det∂β_{N-1}M1, the example should involve K_2^2 rather than K_1^2; please clarify the indexing.
[Section III, Eq. (18)] The notation Rd_{N-1} = overline{Rs_{N-1}} is introduced but the complex conjugation is not clear in the typeset text. Please ensure the overline/conjugation is explicit.
[Appendix A] The appendix lists only N up to 3, while the text refers to Appendix A for details up to N=7. The data files are in supplementary material, but a cross-reference with file names and sizes would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the recurrence (19) is an algebraic claim about known Wronskians, not a fit; the unproved N≥5 case is a correctness gap, not circular structure.

full rationale

The derivation chain starts from the Wronskian representation (9), N_N = det M3, D_N = det M1, taken from Gaillard [12], and from the known failure of the nonlinear superposition formula [2]. The core recurrence (19) is presented as an elimination result: "When one eliminates the two parameters ... it turns out that the result NrN (resp. DrN) is the square of NN−1 (resp. DN−1)". This is an algebraic identity claimed for the determinants, not a parameter fitted to a target, and the arbitrary parameters a_j,b_j remain free variables of the solution. I cannot exhibit any equation in which the claimed output is defined in terms of the same output, nor any fitted constant renamed as a prediction. The main weakness is that (19) is asserted without a general proof and explicitly checked only for N=2,3,4; hence the N=7 expressions inheriting (19) are not independently verified against the NLS residual. That is an omitted proof / correctness risk, not a circularity, because the recurrence and the determinant representation are not equivalent by construction unless the identity holds. No self-citation is load-bearing: the Wronskian input is from an external author [12] and the NLSF failure from [2]. The paper is therefore not circular; score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical parameters are fitted to data: the a_j,b_j are arbitrary variables of the solutions, not fitted constants. The construction assumes the Wronskian representation from reference [12], uses standard block-determinant algebra, and critically assumes the unproved elimination identity Eq. (19) with its polynomiality assertion. No new physical entities are invented.

assumptions (4)

domain assumption The N-th rational rogue wave has the Wronskian quotient representation u_N = det M3 / det M1 from reference [12], with 2N−2 real parameters.
The paper uses this representation as its starting point without proof; the entire recurrence operates on these matrices (Section II, Eq. 9).
standard math Block determinant reduction det [[A,B],[C,D]] = det A · det(D − C A^{-1}B) for nonsingular upper-triangular A.
Standard linear algebra used in Section II (Eq. 13) to lower determinant order.
ad hoc to paper The elimination identities (19) hold for all N, and D_{N−1}^2/D_{N−2}, N_{N−1}^2/N_{N−2} are polynomials.
Asserted in Section III ('it turns out'), illustrated for N=2–4, but no general derivation is supplied. This is the paper-specific load-bearing assumption.
domain assumption The normalization conditions N_N(0,0)=(-1)^N(2N+1), D_N(0,0)=1 define the same solutions.
Needed to fix the arbitrary scale of the quotient; taken from prior literature (Eq. 8).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Explicit higher order rational rogue waves of the nonlinear Schr\"odinger equation." pith.science (2026). https://pith.science/paper/M34BL27U

@misc{pith2026260723151,
  author       = {Pith},
  title        = {Pith review of: Explicit higher order rational rogue waves of the nonlinear Schr\"odinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M34BL27U}},
  note         = {Machine review of arXiv:2607.23151}
}

abstract

The $N$-th order rational rogue wave of the nonlinear Schr\"odinger equation (NLS), which depends on $2 N-2$ real parameters, has been shown to be impossible to generate by the nonlinear superposition formula. We here generate this sequence by a three-term recurrence relation, each step only requiring the computation of three $N-1$-th order determinants of $2 N-2$ variables, while the previous method requires two determinants of order $2 N$ in $2 N$ variables. This allows us to obtain explicitly the seventh wave with its six arbitrary complex parameters. These very compact expressions open the possibility to investigate the possible existence of new patterns in addition to the already observed ones (concentric rings, polygonal configurations, \dots).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 14 canonical work pages

[1]

Adler and R.I

V.\'E. Adler and R.I. Yamilov, Explicit auto-transformations of integrable chains, Journal of Physics A: Mathematical and General 27:2 (1994) 477--492. https://doi.org/10.1088/0305-4470/27/2/030

work page doi:10.1088/0305-4470/27/2/030 1994
[2]

Akhmediev, A

N. Akhmediev, A. Ankiewicz, and J.M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schr\"odinger equation, Phys. Rev. E 80 (2009) 026601 (9pp). https://doi.org/10.1103/PhysRevE.80.026601

work page doi:10.1103/physreve.80.026601 2009
[3]

Akhmediev, V.M

N.N. Akhmediev, V.M. Eleonskii and N.E. Kulagin, Exact first-order solutions of the nonlinear Schr\"odinger equation, [English : Theor. and Math. Phys. 72 (1987) 809--818]. https://doi.org/10.1007/BF01017105

work page doi:10.1007/bf01017105 1987
[4]

https://doi.org/10.12387/C1988052

Chen Deng-yuan, Li Yi-shen and Zeng Yun-bo, The nonlinear superposition formulae for the Zakharov-Shabat eigenvalue problem and the AKNS hierarchy of equations, Acta Mathematicae Applicatae Sinica 11(4) (1988) 468--477. https://doi.org/10.12387/C1988052

work page doi:10.12387/c1988052 1988
[5]

Hsing-Hen Chen, General derivation of B\"acklund transformations from inverse scattering problems, Phys. Rev. Lett. 33 (1974) 925--928. https://doi.org/10.1103/PhysRevLett.33.925

work page doi:10.1103/physrevlett.33.925 1974
[6]

Conte and M

R. Conte and M. Musette, Beyond the two--singular manifold method, Nonlinear physics: theory and experiment , 67--74, eds. E. Alfinito, M. Boiti, L. Martina and F. Pempinelli (World Scientific, Singapore, 1996)

1996
[7]

https://theses.hal.science/tel-00625446/document

Philippe Dubard, Multi-rogue solutions to the focusing NLS equation, Th\`ese, Universit\'e de Bourgogne, Dijon, France (2010). https://theses.hal.science/tel-00625446/document

2010
[8]

Dubard, P

P. Dubard, P. Gaillard, C. Klein and V.B. Matveev, On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation, Eur. Phys. J. Special Topics 185 (2010) 247--258. https://doi.org/10.1140/epjst/e2010-01252-9

work page doi:10.1140/epjst/e2010-01252-9 2010

Show all 23 references

[9]

Dubard and V.B

P. Dubard and V.B. Matveev, Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation Natural hazards and earth system sciences 11 (2011) 667--672. https://doi.org/10.5194/nhess-11-667-2011

2011 doi
[10]

Dubard and V.B

P. Dubard and V.B. Matveev, Multi-rogue waves solutions: from the NLS to the KP-I equation, Nonlinearity 26:12 (2013) R93--R125. https://doi.org/10.1088/0951-7715/26/12/R93 http://stacks.iop.org/Non/26/R93/mmedia (Online movies)

2013 doi
[11]

Eleonskii, I.M

V.M. Eleonskii, I.M. Krichever and N.E. Kulagin, Rational multisoliton solutions of the nonlinear Schro\"odinger equation, Dokl. Akad. Nauk SSSR 287:3 (1986) 606--610. https://www.mathnet.ru/eng/dan47417

1986
[12]

Pierre Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, J. Phys. A: Math. Theor. 44 (2011) 435204 (15pp). https://doi.org/10.1088/1751-8113/44/43/435204

2011 doi
[13]

https://hal.science/hal-00819359/document (14050 pages)

Pierre Gaillard, The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation, preprint (30 April 2013). https://hal.science/hal-00819359/document (14050 pages)

2013
[14]

Pierre Gaillard, Ten-parameter deformations of the sixth-order Peregrine breather solutions of the NLS equation, Phys. Scr. 89 (2014) 015004 (6pp). https://doi.org/10.1088/0031-8949/89/01/015004

2014 doi
[15]

https://doi.org/10.1016/j.aop.2015.01.027 https://ars.els-cdn.com/content/image/1-s2.0-S0003491615000305-mmc1.pdf supplementary material (42 pages)

Pierre Gaillard, Tenth Peregrine breather solution to the NLS equation, Annals of Physics 355 (2015) 293--298. https://doi.org/10.1016/j.aop.2015.01.027 https://ars.els-cdn.com/content/image/1-s2.0-S0003491615000305-mmc1.pdf supplementary material (42 pages)

2015 doi
[16]

https://hal.science/hal-01492325/ https://hal.science/hal-01492325v1/file/hal

Pierre Gaillard and Micka\"el Gastineau, Families of deformations of the thirteen Peregrine breather solutions to the NLS equation depending on twenty-four parameters, Journal of basic and applied research international 21(3) (2017) 130--139. https://hal.science/hal-01492325/ ...

2017
[17]

Its, A.V

A.R. Its, A.V. Rybin, M.A. Sall', Exact integration of nonlinear Schro\"odinger equation, Theoretical and Math. Phys. 74:1 (1988) 20--32. https://doi.org/10.1007/BF01018207 EN

1988 doi
[18]

Lamb Jr, B\"acklund transformations for certain nonlinear evolution equations, J

G.L. Lamb Jr, B\"acklund transformations for certain nonlinear evolution equations, J. Math. Phys. 15 (1974) 2157--2165. https://doi.org/10.1063/1.1666595

1974 doi
[19]

Yasuhiro Ohta and Jianke Yang, General high-order rogue waves and their dynamics in the nonlinear Schr\"odinger equation, Proc. R. Soc. A 168 (2012) 1716--1740. http://dx.doi.org/10.1098/rspa.2011.0640

2012
[20]

Peregrine, Water waves, nonlinear Schr\"odinger equations and their solutions, J

D.H. Peregrine, Water waves, nonlinear Schr\"odinger equations and their solutions, J. Austral. Math. Soc. B 25 (1983) 16--43. Water waves, nonlinear Schrödinger equations and their solutions https://doi.org/10.1017/S0334270000003891

1983 doi
[21]

Rogers and W.F

C. Rogers and W.F. Shadwick, B\"acklund transformations and their applications (Academic press, New York, 1982)

1982
[22]

Xiao Yi, An explicit nonlinear superposition formula for the nonlinear Schr\"odinger equation, Commun. Theor. Phys. 15:2 (1991) 255--256. https://ctp.itp.ac.cn/EN/Y1991/V15/I2/255

1991
[23]

nonlinear sci

Guangxiong Zhang, Peng Huang, Bao-Feng Feng and Chengfa Wu, Rogue waves and their patterns in the vector nonlinear Schr\"odinger equation, J. nonlinear sci. 33:116 (2023) 1--64. https://doi.org/10.1007/s00332-023-09971-5

2023 doi

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Adler and R.I

V.\'E. Adler and R.I. Yamilov, Explicit auto-transformations of integrable chains, Journal of Physics A: Mathematical and General 27:2 (1994) 477--492. https://doi.org/10.1088/0305-4470/27/2/030

work page doi:10.1088/0305-4470/27/2/030 1994

[2] [2]

Akhmediev, A

N. Akhmediev, A. Ankiewicz, and J.M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schr\"odinger equation, Phys. Rev. E 80 (2009) 026601 (9pp). https://doi.org/10.1103/PhysRevE.80.026601

work page doi:10.1103/physreve.80.026601 2009

[3] [3]

Akhmediev, V.M

N.N. Akhmediev, V.M. Eleonskii and N.E. Kulagin, Exact first-order solutions of the nonlinear Schr\"odinger equation, [English : Theor. and Math. Phys. 72 (1987) 809--818]. https://doi.org/10.1007/BF01017105

work page doi:10.1007/bf01017105 1987

[4] [4]

https://doi.org/10.12387/C1988052

Chen Deng-yuan, Li Yi-shen and Zeng Yun-bo, The nonlinear superposition formulae for the Zakharov-Shabat eigenvalue problem and the AKNS hierarchy of equations, Acta Mathematicae Applicatae Sinica 11(4) (1988) 468--477. https://doi.org/10.12387/C1988052

work page doi:10.12387/c1988052 1988

[5] [5]

Hsing-Hen Chen, General derivation of B\"acklund transformations from inverse scattering problems, Phys. Rev. Lett. 33 (1974) 925--928. https://doi.org/10.1103/PhysRevLett.33.925

work page doi:10.1103/physrevlett.33.925 1974

[6] [6]

Conte and M

R. Conte and M. Musette, Beyond the two--singular manifold method, Nonlinear physics: theory and experiment , 67--74, eds. E. Alfinito, M. Boiti, L. Martina and F. Pempinelli (World Scientific, Singapore, 1996)

1996

[7] [7]

https://theses.hal.science/tel-00625446/document

Philippe Dubard, Multi-rogue solutions to the focusing NLS equation, Th\`ese, Universit\'e de Bourgogne, Dijon, France (2010). https://theses.hal.science/tel-00625446/document

2010

[8] [8]

Dubard, P

P. Dubard, P. Gaillard, C. Klein and V.B. Matveev, On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation, Eur. Phys. J. Special Topics 185 (2010) 247--258. https://doi.org/10.1140/epjst/e2010-01252-9

work page doi:10.1140/epjst/e2010-01252-9 2010