arxiv: 2604.23275 · v1 · submitted 2026-04-25 · 🌊 nlin.SI · nlin.PS

Recognition: unknown

Rogue-wave and lump patterns associated with the third Painlev\'{e} equation

Bo Yang , Jianke Yang

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:54 UTC · model grok-4.3

classification 🌊 nlin.SI nlin.PS

keywords rogue wavesUmemura polynomialsthird Painlevé equationlump solutionsnonlinear Schrödinger equationKadomtsev-Petviashvili equationasymptotic patternsintegrable systems

0 comments

The pith

Rogue wave and lump patterns in integrable equations are predicted by roots of Umemura polynomials from the third Painlevé equation

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

The paper shows that in integrable systems such as the nonlinear Schrödinger equation and the Boussinesq equation, rogue-wave patterns become asymptotically predictable from the root distributions of Umemura polynomials once the internal parameters grow large and take a specific form. Simple roots of these polynomials locate fundamental rogue waves through a linear relation in the space-time plane, while multiple roots produce non-fundamental rogue waves clustered near the origin. The same root-based prediction governs lump patterns in a class of higher-order solutions to the Kadomtsev-Petviashvili-I equation at fixed time, again distinguishing fundamental and non-fundamental lumps. The work also supplies a new transformation that converts bilinear rogue-wave solutions of the nonlinear Schrödinger equation into higher-order KPI lumps, underscoring the organizing role of the third Painlevé equation for nonlinear wave structures.

Core claim

In many integrable equations such as the nonlinear Schrödinger equation and the Boussinesq equation, when internal parameters of their rogue wave solutions are large and of certain form, their rogue patterns in the spatial-temporal plane can be asymptotically predicted by root distributions of Umemura polynomials or equivalently pole distributions of rational solutions to the third Painlevé equation. Every simple root induces a fundamental rogue wave whose spatial-temporal location is linearly related to that root, while a multiple root induces a non-fundamental rogue wave in the O(1) neighborhood of the origin. In a certain class of higher-order lump solutions of the Kadomtsev-PetviashviliI

What carries the argument

Root distributions of Umemura polynomials, which map simple roots to linearly offset fundamental waves or lumps and multiple roots to non-fundamental structures near the origin

If this is right

Simple roots of an Umemura polynomial produce fundamental rogue waves whose space-time locations are given by a linear function of the root value
Multiple roots produce non-fundamental rogue waves confined to an O(1) neighborhood of the origin
The same distinction between simple and multiple roots determines whether lumps in KPI solutions are fundamental or non-fundamental at fixed time
The third Painlevé equation supplies the underlying structure that organizes these asymptotic patterns across different integrable models
Bilinear rogue-wave solutions of the nonlinear Schrödinger equation can be mapped directly onto higher-order lump solutions of the KPI equation

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same root-based prediction may extend to rogue-wave or lump solutions in additional integrable or near-integrable models beyond those treated here
The explicit transformation between NLS rogue waves and KPI lumps offers a practical route for constructing new exact solutions in one system from known solutions in the other
Numerical checks of pattern locations against Umemura roots for successively larger parameters can quantify how rapidly the asymptotic regime is approached

Load-bearing premise

Internal parameters must be large and take a specific form for the root distributions of Umemura polynomials to accurately predict the wave and lump patterns

What would settle it

Direct numerical computation of a rogue-wave solution for large internal parameters, followed by checking whether observed wave positions match the linear relation to the corresponding Umemura polynomial roots; systematic mismatch as parameters increase would falsify the asymptotic prediction

Figures

Figures reproduced from arXiv: 2604.23275 by Bo Yang, Jianke Yang.

**Figure 1.** Figure 1: FIG. 1: Root structures of Umemura polynomials view at source ↗

**Figure 2.** Figure 2: FIG. 2: NLS rogue waves view at source ↗

**Figure 3.** Figure 3: FIG. 3: Comparisons between true NLS rogue solutions view at source ↗

**Figure 4.** Figure 4: FIG. 4: Comparisons between true Boussinesq rogue solutions view at source ↗

**Figure 5.** Figure 5: FIG. 5: Predicted KPI lump solutions view at source ↗

**Figure 6.** Figure 6: FIG. 6: True KPI lump solutions view at source ↗

**Figure 7.** Figure 7: FIG. 7: True KPI lump solutions view at source ↗

**Figure 8.** Figure 8: FIG. 8: Quantitative confirmation of Theorem view at source ↗

read the original abstract

We report rogue-wave and lump patterns associated with Umemura polynomials, which arise in rational solutions of the third Painlev\'{e} equation. We first show that in many integrable equations such as the nonlinear Schr\"odinger equation and the Boussinesq equation, when internal parameters of their rogue wave solutions are large and of certain form, then their rogue patterns in the spatial-temporal plane can be asymptotically predicted by root distributions of Umemura polynomials (or equivalently, pole distributions of rational solutions to the third Painlev\'{e} equation). Specifically, every simple root of the Umemura polynomial would induce a fundamental rogue wave whose spatial-temporal location is linearly related to that simple root, while a multiple root of the Umemura polynomial would induce a non-fundamental rogue wave in the $O(1)$ neighborhood of the spatial-temporal origin. Next, we show that in a certain class of higher-order lump solutions of the Kadomtsev-Petviashvili-I (KPI) equation, when their internal parameters are large and of certain form, then their lump patterns at $O(1)$ time can also be predicted asymptotically by root distributions of Umemura polynomials, where simple and multiple roots of the polynomial would give rise to fundamental and non-fundamental lumps in the spatial plane, respectively. These results reveal the importance of the third Painlev\'{e} equation in studies of nonlinear wave patterns. We also report a new transformation which turns bilinear rogue-wave solutions of the nonlinear Schr\"odinger equation to higher-order lump solutions of the KPI equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows rogue-wave and lump locations in NLS, Boussinesq, and KPI are asymptotically given by Umemura polynomial roots for large parameters of specific form, plus a new bilinear map from NLS rogue waves to KPI lumps.

read the letter

The main thing to know is that this work gives an organizing rule for rogue-wave and lump patterns using roots of Umemura polynomials from the third Painlevé equation. When the internal parameters grow large in the right way, simple roots place fundamental waves or lumps at positions linear in those roots, and multiple roots produce a single non-fundamental structure near the origin. They also supply a new transformation that converts bilinear rogue-wave solutions of NLS into higher-order lump solutions of KPI. These are concrete results that extend earlier pattern studies in integrable systems. The constructions for NLS, Boussinesq, and KPI are explicit and the correspondence is checked on the level of the determinant or bilinear representations, which is the part that works cleanly. The link to Painlevé rational solutions is a natural extension of known connections between integrable equations and special polynomials. The soft spot is the asymptotic control. The paper states that the patterns are predicted by the root distributions, but the argument relies on the cross terms becoming negligible in the large-parameter limit. The stress-test concern about possible O(1) interaction shifts is worth checking: if those remainders do not vanish uniformly across the plane, the linear mapping from roots to locations would need a correction term. The abstract and the reported families do not include uniform error bounds, so that part of the justification is thinner than the pattern examples themselves. This paper is for people already working on rogue waves, lumps, and Painlevé transcendents in nonlinear wave equations. A reader who needs concrete examples of how Umemura roots organize multi-wave structures will find usable material here. It is not a broad survey, but the specific claims are sharp enough to merit referee time. I would send it to peer review so the error estimates and the new transformation can be examined in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that, for rogue-wave solutions of the nonlinear Schrödinger and Boussinesq equations and for a class of higher-order lump solutions of the KPI equation, when internal parameters are taken large in a specific form, the resulting spatial-temporal (or spatial) patterns are asymptotically determined by the root distributions of Umemura polynomials (equivalently, poles of rational solutions to the third Painlevé equation). Simple roots produce fundamental rogue waves or lumps whose locations are linearly related to the roots, while multiple roots produce non-fundamental structures near the origin. The paper also presents a new transformation that converts bilinear rogue-wave solutions of the NLS equation into higher-order lump solutions of the KPI equation.

Significance. If the asymptotic claims are supported by uniform error estimates, the work would establish a concrete link between the third Painlevé equation and the large-parameter patterns of rogue waves and lumps across several integrable systems. The explicit constructions for NLS, Boussinesq, and KPI, together with the new transformation, would provide a useful organizing principle for understanding high-order nonlinear wave structures.

major comments (2)

[asymptotic analysis sections for NLS and Boussinesq] The central asymptotic claim (stated in the abstract and developed in the constructions for NLS and Boussinesq) requires that all cross-interaction terms in the underlying determinant or bilinear form become negligible in the large-parameter limit. No uniform o(1) estimate for the remainder term in the plane is supplied; without it, O(1) shifts between rogue waves could persist and invalidate the direct linear mapping to Umemura roots.
[KPI lump section] For the KPI lump patterns at O(1) time, the same large-parameter limit is invoked, yet the manuscript does not demonstrate that the error after the leading root-determined placement remains uniformly small across the spatial plane when multiple roots are present.

minor comments (2)

Notation for the internal parameters and the precise scaling that makes them 'large and of certain form' should be stated explicitly at the beginning of each construction rather than introduced piecemeal.
The new transformation between NLS rogue waves and KPI lumps is announced in the abstract but its precise statement and verification appear only at the end; moving a concise statement of the transformation to an earlier section would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We appreciate the emphasis on the need for rigorous uniform error estimates in the asymptotic analysis. Below, we provide point-by-point responses to the major comments and outline the revisions we will make.

read point-by-point responses

Referee: [asymptotic analysis sections for NLS and Boussinesq] The central asymptotic claim (stated in the abstract and developed in the constructions for NLS and Boussinesq) requires that all cross-interaction terms in the underlying determinant or bilinear form become negligible in the large-parameter limit. No uniform o(1) estimate for the remainder term in the plane is supplied; without it, O(1) shifts between rogue waves could persist and invalidate the direct linear mapping to Umemura roots.

Authors: We agree that a uniform o(1) estimate for the remainder is essential to fully justify the asymptotic mapping without possible persistent shifts. In the current version, we have derived the leading-order terms by analyzing the dominant contributions in the determinant expressions when parameters are large, showing that the positions align linearly with the roots for simple roots and cluster near the origin for multiples. To address this, we will include additional analysis providing uniform bounds on the error terms across the entire plane. This will involve detailed estimates on the off-diagonal terms in the bilinear forms and the use of the root separation properties of Umemura polynomials. We believe this can be done rigorously and will strengthen the paper. revision: yes
Referee: [KPI lump section] For the KPI lump patterns at O(1) time, the same large-parameter limit is invoked, yet the manuscript does not demonstrate that the error after the leading root-determined placement remains uniformly small across the spatial plane when multiple roots are present.

Authors: We acknowledge the validity of this observation. The manuscript currently focuses on the leading asymptotic placement for both simple and multiple roots in the KPI lumps at fixed time. For multiple roots, the non-fundamental structures are described near the origin, but uniform smallness of the error needs explicit demonstration. In the revision, we will add uniform error estimates for the KPI case, extending the bilinear analysis to show that the remainder is o(1) uniformly in space. This will cover the scenarios with multiple roots by considering the local behavior near the origin separately if necessary. revision: yes

Circularity Check

0 steps flagged

No significant circularity; asymptotic derivations are self-contained

full rationale

The paper derives rogue-wave and lump patterns as asymptotic limits of explicit known solutions (bilinear/determinant forms) for NLS, Boussinesq, and KPI when internal parameters become large in specified forms. These limits are mapped to root distributions of Umemura polynomials, which are independently defined via rational solutions of the third Painlevé equation. No step equates a prediction to its input by construction, renames a fitted quantity, or relies on a self-citation chain for the central claim. The results are presented as new observations from existing solution families rather than tautologies.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identified; the work relies on standard asymptotic analysis of known integrable solutions.

pith-pipeline@v0.9.0 · 5590 in / 1183 out tokens · 37954 ms · 2026-05-08T06:54:45.854847+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

95 extracted references

[1]

,±(N−1), thenU N(z;µ)has a zero root of multiplicityN 0(N0 + 1)/2, where N0 =N− |µ|

ifµis equal to one of0,±1, . . . ,±(N−1), thenU N(z;µ)has a zero root of multiplicityN 0(N0 + 1)/2, where N0 =N− |µ|. The nonzero roots are all simple, and their number isN p =N(N+ 1)/2−N 0(N0 + 1)/2
[2]

,±(N−1), thenU N(z;µ)hasN(N+ 1)/2nonzero simple roots only

ifµ̸= 0,±1, . . . ,±(N−1), thenU N(z;µ)hasN(N+ 1)/2nonzero simple roots only. This lemma gives very clear results regarding multiplicities of the zero and nonzero roots in Umemura polynomi- als. These clear root results will lead to clear rogue-wave and lump pattern predictions associated with Umemura polynomials that we will derive later in this paper. F...
[3]

Ifµis equal to one of0,±1, . . . ,±(N−1), then the rogue waveu N(x, t)asymptotically splits intoN p fundamental (Peregrine) rogue waves located far away from the spatial-temporal origin, plus aN 0-th order super rogue wave in theO(1)neighborhood of the origin, whereN p andN 0 are as given in Lemma 1. These Peregrine waves are ˆu1(x−ˆx0, t− ˆt0)e it, where...
[4]

,±(N−1), then the rogue waveu N(x, t)asymptotically splits intoN(N+ 1)/2Peregrine waves ˆu1(x−ˆx 0, t− ˆt0)e it, whose spatial-temporal locations(ˆx 0, ˆt0)are given by Eq

Ifµ̸= 0,±1, . . . ,±(N−1), then the rogue waveu N(x, t)asymptotically splits intoN(N+ 1)/2Peregrine waves ˆu1(x−ˆx 0, t− ˆt0)e it, whose spatial-temporal locations(ˆx 0, ˆt0)are given by Eq. (3.15), withz 0 being each of theN(N+ 1)/2simple nonzero roots of the Umemura polynomialU N(z;µ). The error of this Peregrine wave approximation isO(|A| −1). This the...
[5]

Ifµis equal to one of0,±1, . . . ,±(N−1), then the rogue waveu N(x, t)asymptotically splits intoN p funda- mental rogue waves located far away from the spatial-temporal origin, plus aN 0-th order rogue wave in theO(1) neighborhood of the origin, whereN p andN 0 are as given in Lemma 1. These fundamental rogue waves are u1(x−ˆx0, t− ˆt0), whereu 1(x, t)is ...
[6]

,±(N−1), then the rogue waveu N(x, t)asymptotically splits intoN(N+ 1)/2fundamental rogue wavesu 1(x−ˆx0, t− ˆt0), whose spatial-temporal locations(ˆx0, ˆt0)are given by Eq

Ifµ̸= 0,±1, . . . ,±(N−1), then the rogue waveu N(x, t)asymptotically splits intoN(N+ 1)/2fundamental rogue wavesu 1(x−ˆx0, t− ˆt0), whose spatial-temporal locations(ˆx0, ˆt0)are given by Eq. (3.42), withz 0 being each of theN(N+ 1)/2simple nonzero roots of the Umemura polynomialU N(z;µ). The error of this fundamental rogue wave approximation isO(|A| −1)....
[7]

Ifµis equal to one of0,±1, . . . ,±(N−1), then the wave field asymptotically splits intoN p fundamental lumps located far away from the lump center(x 0, y0) = (12t,0), plus aN 0-th order super lump in theO(1)neighborhood of this lump center, whereN p andN 0 are as given in Lemma 1. These fundamental lumps areu 1(x−ˆx0, y−ˆy0, t), whereu 1(x, y, t)is given...
[8]

,±(N−1), then the the wave field asymptotically splits intoN(N+ 1)/2fundamental lumps u1(x−ˆx0, t− ˆt0), where(ˆx0, ˆt0)are given by Eq

Ifµ̸= 0,±1, . . . ,±(N−1), then the the wave field asymptotically splits intoN(N+ 1)/2fundamental lumps u1(x−ˆx0, t− ˆt0), where(ˆx0, ˆt0)are given by Eq. (4.16), withz 0 being each of theN(N+ 1)/2simple nonzero roots of the Umemura polynomialU N(z;µ). The error of this fundamental-lump approximation isO(|A| −1). Proof.We first rewrite the determinant ofσ...
[9]

The (ˆx, y) intervals are−70≤ˆx≤50,−30≤y≤30 for the upper row, and−60≤ˆx≤60,−30≤y≤30 for the lower row, where ˆx≡x−12tis the movingx-coordinate

Upper row:A= 10,µ= 2; lower row:A= 20,µ= 1/100. The (ˆx, y) intervals are−70≤ˆx≤50,−30≤y≤30 for the upper row, and−60≤ˆx≤60,−30≤y≤30 for the lower row, where ˆx≡x−12tis the movingx-coordinate. For the upper middle panel, the color bar does not show the full solution range. To confirm our predictions for the second example in the lower row of Fig. 5, we pl...
[10]

M´ emoire sur les ´ equations diff´ erentielles dont l’int´ egrale g´ en´ erale est uniforme

P. Painlev´ e, “M´ emoire sur les ´ equations diff´ erentielles dont l’int´ egrale g´ en´ erale est uniforme”, Bull. Soc. Math. Fr., 28, 201 (1900)

1900
[11]

Sur les ´ equations diff´ erentielles du second ordre et d’ordre sup´ erieur dont l’int´ egrale g´ en´ erale est uniforme

P. Painlev´ e, “Sur les ´ equations diff´ erentielles du second ordre et d’ordre sup´ erieur dont l’int´ egrale g´ en´ erale est uniforme”, Acta Math., 25, 1 (1902)

1902
[12]

Sur les ´ equations diff´ erentielles du second ordre et du premier degr´ e dont l’int´ egrale g´ en´ erale est ` a points critiques fixes

B. Gambier, “Sur les ´ equations diff´ erentielles du second ordre et du premier degr´ e dont l’int´ egrale g´ en´ erale est ` a points critiques fixes”, Acta Math., 33, 1 (1910)

1910
[13]

Painlev´ e Transcendents

P.A. Clarkson, “Painlev´ e Transcendents”, Chapter 32 of the NIST Digital Library of Mathematical Functions
[14]

Universality for the focusing nonlinear Schr¨ odinger equation at the gradient catastrophe point: rational breathers and poles of the tritronqu´ ee solution to Painlev´ e I

M. Bertola and A. Tobvis, “Universality for the focusing nonlinear Schr¨ odinger equation at the gradient catastrophe point: rational breathers and poles of the tritronqu´ ee solution to Painlev´ e I”, Comm. Pure Appl. Math. 66, 678 (2013)

2013
[15]

The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix,

R. Buckingham and P. D. Miller, “The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix,” Journal d’Analyse Math´ ematique 118, 397 (2012)

2012
[16]

Rogue wave patterns in the nonlinear Schr¨ odinger equation

B. Yang and J. Yang, “Rogue wave patterns in the nonlinear Schr¨ odinger equation”, Physica D 419, 132850 (2021)

2021
[17]

Yang and J

B. Yang and J. Yang,Rogue Waves in Integrable Systems(Springer, New York, 2024)

2024
[18]

Rogue wave patterns associated with Okamoto polynomial hierarchies

B. Yang and J. Yang, “Rogue wave patterns associated with Okamoto polynomial hierarchies”, Stud. Appl. Math. 151, 60 (2023)

2023
[19]

Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation

B. Yang and J. Yang, “Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation”, Phys. Lett. A 458, 128573 (2023)

2023
[20]

Pattern transformation in higher-order lumps of the Kadomtsev–Petviashvili I equation

B. Yang and J. Yang, “Pattern transformation in higher-order lumps of the Kadomtsev–Petviashvili I equation”, J. Nonl. Sci. 32, 52 (2022)

2022
[21]

The fourth Painlev´ e equation and associated special polynomials

P.A. Clarkson, “The fourth Painlev´ e equation and associated special polynomials”, J. Math. Phys. 44, 5350 (2003)

2003
[22]

Reducibility of Painlev´ e equations

V.I. Gromak, “Reducibility of Painlev´ e equations”, Differential Equations 20, 1191 (1984)

1984
[23]

Classical solutions of the third Painlev´ e equation

Y. Murata, “Classical solutions of the third Painlev´ e equation”, Nagoya Math. J. 139, 37 (1995)

1995
[24]

Painlev´ e equations in the past 100 years

H. Umemura, “Painlev´ e equations in the past 100 years”, Trans. Amer. Math. Soc. 204, 81 (2001). 24

2001
[25]

The third Painlev´ e equation and associated special polynomials

P.A. Clarkson, “The third Painlev´ e equation and associated special polynomials”, J. Phys. A 36, 9507 (2003)

2003
[26]

On aq-difference Painlev´ e III equation: II. Rational solutions

K. Kajiwara, “On aq-difference Painlev´ e III equation: II. Rational solutions”, J. Nonlinear Math. Phys. 10, 282 (2003)

2003
[27]

On the Umemura polynomials for the Painlev´ e III equation

K. Kajiwara and T. Masuda, “On the Umemura polynomials for the Painlev´ e III equation”, Phys. Lett. A 260, 462 (1999)

1999
[28]

A constructive proof for the Umemura polynomials of the third Painlev´ e equation

P. A. Clarkson, C. K. Law and C. Lin, “A constructive proof for the Umemura polynomials of the third Painlev´ e equation”, Symmetry Integrability Geom. Methods Appl. 19, 080 (2023)

2023
[29]

Discriminants of Umemura polynomials associated to Painlev´ e III

T. Amdeberhan, “Discriminants of Umemura polynomials associated to Painlev´ e III”, Phys. Lett. A 354, 410 (2006)

2006
[30]

The second Painlev´ e equation, its hierarchy and associated special polynomials

P. A. Clarkson and E. L. Mansfield, “The second Painlev´ e equation, its hierarchy and associated special polynomials”, Nonlinearity 16, R1 (2003)

2003
[31]

Zeros of Wronskians of Hermite polynomials and Young diagrams

G. Felder, A. D. Hemery and A.P. Veselov, “Zeros of Wronskians of Hermite polynomials and Young diagrams”, Physica D 241, 2131 (2012)

2012
[32]

Rational solutions of the Painlev´ e-III equation

T. Bothner, P.D. Miller and Y. Sheng, “Rational solutions of the Painlev´ e-III equation”, Stud. Appl. Math. 141, 626 (2018)

2018
[33]

Rational solutions of the Painlev´ e-III equation: large parameter asymptotics

T. Bothner and P.D. Miller, “Rational solutions of the Painlev´ e-III equation: large parameter asymptotics”, Constr. Approx. 51, 123 (2020)

2020
[34]

On a class of polynomials associated with the Korteweg de Vries equation

M. Adler and J. Moser, “On a class of polynomials associated with the Korteweg de Vries equation”, Commun. Math. Phys. 61, 1 (1978)

1978
[35]

Vortices and polynomials

P. A. Clarkson, “Vortices and polynomials”, Stud. Appl. Math. 123, 37 (2009)

2009
[36]

The propagation of nonlinear wave envelopes

D. J. Benney and A. C. Newell, “The propagation of nonlinear wave envelopes”, J. Math. Phys. 46, 133 (1967)

1967
[37]

Stability of periodic waves of finite amplitude on the surface of a deep fluid,

V.E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid,” Zh. Prikl. Mekh. Tekh. Fiz 9, 86 (1968) (Transl. in J. Appl. Mech. Tech. Phys. 9, 190)

1968
[38]

Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers”, Appl. Phys. Lett. 23, 142 (1973)

1973
[39]

Water waves, nonlinear Schr¨ odinger equations and their solutions

D.H. Peregrine, “Water waves, nonlinear Schr¨ odinger equations and their solutions”, J. Aust. Math. Soc. B 25, 16 (1983)

1983
[40]

Rogue waves and rational solutions of the nonlinear Schr¨ odinger equation,

N. Akhmediev, A. Ankiewicz and J.M. Soto-Crespo, “Rogue waves and rational solutions of the nonlinear Schr¨ odinger equation,” Phys. Rev. E 80, 026601 (2009)

2009
[41]

On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation

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. Spec. Top. 185, 247 (2010)

2010
[42]

Circular rogue wave clusters

D.J. Kedziora, A. Ankiewicz and N. Akhmediev, “Circular rogue wave clusters”, Phys. Rev. E 84, 056611 (2011)

2011
[43]

Nonlinear Schr¨ odinger equation: generalized Darboux transformation and rogue wave solutions

B.L. Guo, L.M. Ling and Q.P. Liu, “Nonlinear Schr¨ odinger equation: generalized Darboux transformation and rogue wave solutions”, Phys. Rev. E 85, 026607 (2012)

2012
[44]

General high-order rogue waves and their dynamics in the nonlinear Schr¨ odinger equation,

Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schr¨ odinger equation,” Proc. R. Soc. Lond. A 468, 1716 (2012)

2012
[45]

Rogue wave observation in a water wave tank,

A. Chabchoub, N. Hoffmann and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011)

2011
[46]

Observation of a hierarchy of up to fifth-order rogue waves in a water tank,

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012)

2012
[47]

Super rogue waves: observation of a higher-order breather in water waves

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super rogue waves: observation of a higher-order breather in water waves”, Phys. Rev. X 2, 011015 (2012)

2012
[48]

Observation of rogue wave triplets in water waves

A. Chabchoub and N. Akhmediev, “Observation of rogue wave triplets in water waves”, Phys. Lett. A 377, 2590 (2013)

2013
[49]

The Peregrine soliton in nonlinear fibre optics,

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev and J.M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010)

2010
[50]

Higher-order modulation instability in nonlinear fiber optics

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J.M. Dudley and G. Genty, “Higher-order modulation instability in nonlinear fiber optics”, Phys. Rev. Lett. 107, 253901 (2011)

2011
[51]

Observation of Peregrine solitons in a multicomponent plasma with negative ions

H. Bailung, S.K. Sharma and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions”, Phys. Rev. Lett. 107, 255005 (2011)

2011
[52]

Experimental realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates

A. Romero-Ros, G.C. Katsimiga, S.I. Mistakidis, S. Mossman, G. Biondini, P. Schmelcher, P. Engels, and P.G. Kevrekidis, “Experimental realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates”, Phys. Rev. Lett. 132, 033402 (2024)

2024
[53]

Generation of acoustic rogue waves in dusty plasmas through three-dimensional particle focusing by distorted waveforms

Y. Tsai, J. Tsai and L. I, “Generation of acoustic rogue waves in dusty plasmas through three-dimensional particle focusing by distorted waveforms”, Nat. Phys. 12, 573 (2016)

2016
[54]

Rogue waves, rational solutions, the patterns of their zeros and integral relations

A. Ankiewicz, P. Clarkson and N. Akhmediev, “Rogue waves, rational solutions, the patterns of their zeros and integral relations”, J. Phys. A, Math. Theor. 43 (2010) 122002

2010
[55]

The height of an nth-order fundamental rogue wave for the nonlinear Schr¨ odinger equation

L. Wang, C.H. Yang, J. Wang and J.S. He, “The height of an nth-order fundamental rogue wave for the nonlinear Schr¨ odinger equation”, Phys. Lett. A 381, 1714 (2017)

2017
[56]

Extreme superposition: Rogue waves of infinite order and the Painlev´ e-III hierarchy

D. Bilman, L.M. Ling and P.D. Miller, “Extreme superposition: Rogue waves of infinite order and the Painlev´ e-III hierarchy”, Duke Math. J. 169, 671 (2020)

2020
[57]

Classifying the hierarchy of nonlinear-Schr¨ odinger-equation rogue-wave solutions

D. J. Kedziora, A. Ankiewicz and N. Akhmediev, “Classifying the hierarchy of nonlinear-Schr¨ odinger-equation rogue-wave solutions”, Phys. Rev. E 88, 013207 (2013)

2013
[58]

Generating mechanism for higher-order rogue waves

J.S. He, H.R. Zhang, L.H. Wang, K. Porsezian and A.S. Fokas, “Generating mechanism for higher-order rogue waves”, Phys. Rev. E 87, 052914 (2013)

2013
[59]

Rogue wave patterns associated with Adler-Moser polynomials in the nonlinear Schr¨ oodinger equation

B. Yang, J. Yang, “Rogue wave patterns associated with Adler-Moser polynomials in the nonlinear Schr¨ oodinger equation”, Appl. Math. Lett. 148, 108871 (2024)

2024
[60]

Rogue wave patterns associated with Adler–Moser polynomials featuring multiple roots in the nonlinear Schr¨ odinger equation

H. Lin and L.M. Ling, “Rogue wave patterns associated with Adler–Moser polynomials featuring multiple roots in the nonlinear Schr¨ odinger equation”, Stud. Appl. Math. 154, e12782 (2025). 25

2025
[61]

Triangular rogue clusters associated with multiple roots of Adler-Moser polynomials in integrable systems

B. Yang and J. Yang, “Triangular rogue clusters associated with multiple roots of Adler-Moser polynomials in integrable systems”, Physica D 483, 134921 (2025)

2025
[62]

Universal rogue wave patterns associated with the Yablonskii-Vorob’ev polynomial hierarchy

B. Yang and J. Yang, “Universal rogue wave patterns associated with the Yablonskii-Vorob’ev polynomial hierarchy”, Physica D 425, 132958 (2021)

2021
[63]

Th´ eorie de I’intumescence Liquid, Appleteonde Solitaire au de Translation, se Propageantdansun Canal Rectangulaire

J. Boussinesq, “Th´ eorie de I’intumescence Liquid, Appleteonde Solitaire au de Translation, se Propageantdansun Canal Rectangulaire”, Comptes Rendus, 72, 755 (1871)
[64]

J. Boussinesq, “Th´ eorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en commu- niquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond”, J. Pure Appl. 17, 55 (1872)
[65]

Rational solutions of the Boussinesq equation and applications to rogue waves

P. A. Clarkson and E. Dowie, “Rational solutions of the Boussinesq equation and applications to rogue waves”, Trans. Math. Appl. 1, 1 (2017)

2017
[66]

Rational growing mode: exact solutions to the Boussinesq equation

M. Tajiri and Y. Murakami, “Rational growing mode: exact solutions to the Boussinesq equation”, J. Phys. Soc. Japan 60, 2791 (1991)

1991
[67]

Rogue waves and hybrid solutions of the Boussinesq equation

J. Rao, Y. Liu, C. Qian and J. He, “Rogue waves and hybrid solutions of the Boussinesq equation”, Z. Naturforsch. A 72, 307 (2017)

2017
[68]

General rogue waves in the Boussinesq equation

B. Yang and J. Yang, “General rogue waves in the Boussinesq equation”, J. Phys. Soc. Jpn. 89, 024003 (2020)

2020
[69]

On the stability of solitary waves in weakly dispersive media

B. B. Kadomtsev and V.I. Petviashvili, “On the stability of solitary waves in weakly dispersive media”, Sov. Phys. Dokl. 15, 539 (1970)

1970
[70]

On the evolution of packets of water waves

M. J. Ablowitz and H. Segur, “On the evolution of packets of water waves”, J. Fluid Mech. 92, 691 (1979)

1979
[71]

Self-focusing of plane dark solitons in nonlinear defocusing media

D. E. Pelinovsky, Yu. A. Stepanyants and Yu.A. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media”, Phys. Rev. E 51, 5016 (1995)

1995
[72]

Optical kerr spatiotemporal dark-lump dynamics of hydrodynamic origin

F. Baronio, S. Wabnitz and Y. Kodama, “Optical kerr spatiotemporal dark-lump dynamics of hydrodynamic origin”, Phys. Rev. Lett. 116, 173901 (2016)

2016
[73]

Observation of lump solitons

L. Dieli, D. Pierangeli, F. Baronio, S. Trillo and C. Conti, “Observation of lump solitons”, Phys. Rev. Lett. 136, 053804 (2026)

2026
[74]

Soliton-like bubbles in the system of interacting bosons

I.V. Barashenkov and V.G. Makhankov, “Soliton-like bubbles in the system of interacting bosons”, Phys. Lett. A 128, 52 (1988)

1988
[75]

Solitons in two-dimensional Bose-Einstein condensates

S. Tsuchiya, F. Dalfovo and L. P. Pitaevskii, “Solitons in two-dimensional Bose-Einstein condensates”, Phys. Rev. A 77, 045601 (2008)

2008
[76]

Hydrody- namics and two-dimensional dark lump solitons for polariton superfluids

D.J. Frantzeskakis, T. Horikis, A.S. Rodrigues, P. Kevrekidis, R. Carretero-Gonz´ alez and J. Cuevas-Maraver, “Hydrody- namics and two-dimensional dark lump solitons for polariton superfluids”, Phys. Rev. E 98, 022205 (2018)

2018
[77]

Novikov, S.V

S. Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov,Theory of Solitons: the Inverse Scattering Method(Plenum, 1984)

1984
[78]

Ablowitz and P.A

M.J. Ablowitz and P.A. Clarkson,Solitons, Nonlinear Evolution Equations and Inverse Scattering(Cambridge University Press, 1991)

1991
[79]

Equation of an extraordinary soliton

V.I. Petviashvili, “Equation of an extraordinary soliton”, Plasma Physics 2, 469 (1976)

1976
[80]

Two-dimensional solitons of the Kadomtsev– Petviashvili equation and their interaction

S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its and V.B. Matveev, “Two-dimensional solitons of the Kadomtsev– Petviashvili equation and their interaction”, Phys. Lett. A 63, 205 (1977)

1977

Showing first 80 references.