Rogue waves for semilinear wave equations

Julia Henninger

arxiv: 2606.07021 · v1 · pith:R7G7JRUZnew · submitted 2026-06-05 · 🧮 math.AP

Rogue waves for semilinear wave equations

Julia Henninger This is my paper

Pith reviewed 2026-06-27 21:43 UTC · model grok-4.3

classification 🧮 math.AP

keywords rogue wavessemilinear wave equationsvariational methodscritical pointslocalized solutionsspectral propertieswave operator

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The pith

Semilinear wave equations admit rogue wave solutions localized in space and time under spectral conditions on coefficients.

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

The paper establishes existence of solutions localized in both space and time for the semilinear wave equation with variable coefficients. These rogue waves arise as critical points of an energy functional on a Hilbert space. The argument requires sufficient conditions on the coefficients V, tilde V, d, tilde d, the elliptic operator M and p greater than 1, derived from the spectral properties of the wave-type operator. Critical points are shown to yield weak solutions after regularity analysis. Concrete examples of coefficients and operators meeting the assumptions are supplied.

Core claim

Under sufficient conditions on the coefficients V, tilde V, d, tilde d, the elliptic operator M and p>1, the energy functional on the Hilbert space possesses a critical point. This critical point corresponds to a weak solution of the equation V(x) partial_t^2 u + d(t) M(x, nabla_x) u = tilde V(x) tilde d(t) |u|^{p-1} u that is localized in space and time. The existence proof relies on a detailed analysis of the spectral properties of the wave-type operator.

What carries the argument

The energy functional on a suitable Hilbert space, whose critical points are located using the spectral properties of the wave-type operator.

If this is right

The equation has weak solutions that are localized in space and time for any coefficients meeting the assumptions.
The critical points obtained are genuine weak solutions after the regularity step is applied.
The assumptions are realizable, as shown by the provided examples of coefficients and elliptic operators.

Where Pith is reading between the lines

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

The same variational construction could be tested on related nonlinear wave models with different nonlinearities.
Stability of the obtained localized solutions under small perturbations remains open for investigation.
Numerical approximation of the energy functional for the example coefficients could provide independent confirmation of the critical point.

Load-bearing premise

The spectral properties of the wave-type operator guarantee that the energy functional has a critical point when the coefficients and p satisfy the stated conditions.

What would settle it

An explicit set of coefficients and operator satisfying the spectral conditions for which no critical point of the energy functional exists would show the claim fails.

read the original abstract

We study the semilinear wave equation $ V(x) \partial_t^2 u + d(t) M(x,\nabla_x) u=\tilde{V}(x) \tilde{d}(t) |u|^{p-1}u$ on $ \mathbb{R}^N \times \mathbb{R}$ and show the existence of solutions which are localized in space and in time, called rogue waves, by means of variational methods. We introduce an energy functional on a suitable Hilbert space, and provide sufficient conditions on the coefficients $V, \tilde{V}, d, \tilde{d}$, the elliptic operator $M$ and $p>1$ for the existence of a critical point. Our approach is based on a detailed analysis of the wave type operator and in particular its spectral properties. Further regularity considerations show that critical points are weak solutions to our equation. Moreover, we provide examples of the coefficients and the elliptic operator which satisfy our assumptions.

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

This paper gives an existence result for rogue-wave solutions to a semilinear wave equation with space- and time-dependent coefficients, using a variational critical-point argument on a Hilbert space.

read the letter

The main takeaway is that the authors prove existence of solutions localized in both space and time for the equation V(x) u_tt + d(t) M(x,∇u) = Vtilde(x) dtilde(t) |u|^{p-1}u, under sufficient conditions on the coefficients, the elliptic operator M, and p>1. They do this by setting up an energy functional whose critical points are the desired solutions, relying on spectral properties of the wave operator.

What stands out is the extension to coefficients that vary in both space and time, plus the inclusion of concrete examples that meet the assumptions. They also add a regularity step to confirm that critical points are weak solutions. That combination is the actual new piece.

The soft spots are modest. The abstract leaves the precise conditions on V, d, M and the spectral gap implicit, so it is difficult to judge how broad or sharp the result really is. The approach follows the usual mountain-pass plus spectral analysis template for semilinear hyperbolic problems; without the full proof it is not clear whether the spectral analysis contains any technical novelty or is mostly routine application. No numerical checks or explicit solutions are mentioned, which is normal for this style of work but means the result stays purely theoretical.

This is for people already working on variational methods for nonlinear wave equations or on rogue-wave constructions in PDEs. A reader in that niche could use the conditions and examples as a reference. The paper is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee even if the broader impact stays inside the subfield. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to prove the existence of rogue-wave solutions (localized in both space and time) to the semilinear wave equation V(x) ∂_t² u + d(t) M(x,∇_x) u = ilde{V}(x) ilde{d}(t) |u|^{p-1}u on \mathbb{R}^N imes \mathbb{R} by variational methods. An energy functional is introduced on a suitable Hilbert space; sufficient conditions on the coefficients V, ilde{V}, d, ilde{d}, the elliptic operator M, and p>1 are stated to guarantee a critical point via spectral properties of the wave-type operator. Critical points are shown to be weak solutions, and concrete examples of coefficients and operators satisfying the assumptions are supplied.

Significance. If the stated sufficient conditions hold and the spectral analysis establishes the required mountain-pass geometry and Palais-Smale condition, the work supplies a variational existence result for time- and space-localized solutions of variable-coefficient semilinear wave equations. The explicit examples increase the result's applicability and verifiability.

minor comments (2)

The precise definition of the Hilbert space and the precise form of the energy functional should be stated in the introduction rather than deferred entirely to later sections, to improve readability.
Notation for the elliptic operator M(x,∇_x) and the time-dependent coefficients should be introduced with a short display equation in §1 to avoid ambiguity when the assumptions are listed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes existence of localized solutions (rogue waves) for the given semilinear wave equation by constructing an energy functional on a Hilbert space whose critical points are weak solutions, under stated sufficient conditions on the coefficients, the elliptic operator M, and p>1. The argument rests on a spectral analysis of the wave-type operator to verify the geometry and Palais-Smale condition needed for a critical-point theorem. This is a direct, self-contained application of standard variational methods; the functional is defined from the PDE itself, the spectral hypotheses are independent external assumptions, and no step reduces by construction to a fitted parameter, self-citation chain, or renaming of the target result. No load-bearing premise collapses into the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a critical point for the energy functional, which depends on unverified spectral properties of the operator and paper-specific sufficient conditions on the coefficients; no free parameters or invented entities are evident from the abstract.

axioms (2)

domain assumption The wave type operator possesses spectral properties that enable the energy functional to satisfy the geometry and compactness conditions for a critical point via variational theorems.
Invoked in the detailed analysis of the wave type operator.
ad hoc to paper The coefficients V, ilde{V}, d, ilde{d} and operator M satisfy the sufficient conditions stated in the paper.
These are the assumptions under which the existence is claimed.

pith-pipeline@v0.9.1-grok · 5677 in / 1336 out tokens · 28708 ms · 2026-06-27T21:43:32.164177+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 14 canonical work pages

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CRC 1173 Preprint 2025/40

Guido Schneider and Nils Thorin.Approximate Peregrine solitons in dispersive nonlinear wave equations. CRC 1173 Preprint 2025/40. Karlsruhe Institute of Technology, July 2025.doi:10.5445/IR/1000183475.url:https://www.waves.kit.edu/downloads/ CRC1173_Preprint_2025-40.pdf

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Theory of electron spin resonance in scanning tunneling mi- croscopy

C. B. Ward et al. “Rogue waves and periodic solutions of a nonlocal nonlinear Schrödinger model”. In:Phys. Rev. Res.2 (1 Mar. 2020), p. 013351.doi:10.1103/PhysRevResearch. 2.013351.url:https://link.aps.org/doi/10.1103/PhysRevResearch.2.013351

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Eigenfunctions of a weighted Laplace operator in the whole space

Nouria Arar and Tahar Z. Boulmezaoud. “Eigenfunctions of a weighted Laplace operator in the whole space”. In:J. Math. Anal. Appl.400.1 (2013), pp. 161–173.issn: 0022- 247X,1096-0813.doi:10.1016/j.jmaa.2012.06.009.url:https://doi.org/10.1016/ j.jmaa.2012.06.009. REFERENCES 31

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Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

Jinbing Chen and Dmitry E. Pelinovsky. “Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background”. In:Journal of Nonlinear Science 29 (6 2019).doi:https://doi.org/10.1007/s00332-019-09559-y

work page doi:10.1007/s00332-019-09559-y 2019

[3] [3]

Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation

Jinbing Chen and Dmitry E. Pelinovsky. “Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation”. In:Phys. Rev. E103 (6 June 2021), p. 062206.doi:10.1103/PhysRevE.103.062206.url:https://link. aps.org/doi/10.1103/PhysRevE.103.062206

work page doi:10.1103/physreve.103.062206.url:https://link 2021

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Periodic standing waves in the focusing nonlinear Schrödinger equation: Rogue waves and modulation instability

Jinbing Chen, Dmitry E. Pelinovsky, and Robert E. White. “Periodic standing waves in the focusing nonlinear Schrödinger equation: Rogue waves and modulation instability”. In:Physica D: Nonlinear Phenomena405 (2020), p. 132378.issn: 0167-2789.doi:https: //doi.org/10.1016/j.physd.2020.132378.url:https://www.sciencedirect.com/ science/article/pii/S0167278919304506

work page doi:10.1016/j.physd.2020.132378.url:https://www.sciencedirect.com/ 2020

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Nocedal, S

Alexey Slunyaev Christian Kharif Efim Pelinovsky.Rogue Waves in the Ocean. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer Berlin, Heidel- berg, 2009, pp. XIV, 216.isbn: 978-3-540-88418-7.doi:https://doi.org/10.1007/978- 3-540-88419-4. [6]NIST Digital Library of Mathematical Functions.https : / / dlmf . nist . gov/, Release 1...

work page doi:10.1007/978- 2009

[6] [6]

Eastham.The Spectral Theory of Periodic Differential Equations

M.S.P. Eastham.The Spectral Theory of Periodic Differential Equations. Texts in math- ematics. Scottish Academic Press [distributed by Chatto & Windus, London, 1973.isbn: 9780701119362.url:https://books.google.de/books?id=LUHvAAAAMAAJ

1973

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Arthur Erdélyi et al.Higher transcendental functions. Vol. II. Based on notes left by Harry Bateman, Reprint of the 1953 original. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1981, pp. xviii+396.isbn: 0-89874-069-X

1953

[8] [8]

Mathematical theory and applications in physics

Boling Guo et al.Rogue waves. Mathematical theory and applications in physics. De Gruyter, Berlin, 2017, pp. vii+204.isbn: 978-3-11-046942-4; 978-3-11-046969-1.doi:10. 1515/9783110470574.url:https://doi.org/10.1515/9783110470574

work page doi:10.1515/9783110470574 2017

[9] [10]

Breather solutions for semilin- ear wave equations

Julia Henninger, Sebastian Ohrem, and Wolfgang Reichel. “Breather solutions for semilin- ear wave equations”. In:Journal of Mathematical Analysis and Applications564.1 (2026), p. 130829.issn: 0022-247X.doi:https://doi.org/10.1016/j.jmaa.2026.130829. url:https://www.sciencedirect.com/science/article/pii/S0022247X26004415

work page doi:10.1016/j.jmaa.2026.130829 2026

[10] [11]

Symmetric tensor spherical harmonics on the N-sphere and their appli- cation to the de Sitter group SO (N, 1)

Atsushi Higuchi. “Symmetric tensor spherical harmonics on the N-sphere and their appli- cation to the de Sitter group SO (N, 1)”. In:Journal of mathematical physics28.7 (1987), pp. 1553–1566

1987

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Tosio Kato.Perturbation theory for linear operators. Vol. Band 132. Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York, Inc., New York, 1966, pp. xix+592

1966

[12] [13]

Inequalities for the Associated Legendre Functions

G. Lohöfer. “Inequalities for the Associated Legendre Functions”. In:Journal of Approx- imation Theory95.2 (1998), pp. 178–193.issn: 0021-9045.doi:https://doi.org/10. 1006/jath.1998.3207.url:https://www.sciencedirect.com/science/article/ pii/S0021904598932075. 32 REFERENCES

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Bociu, B

Daniela Maier, Wolfgang Reichel, and Guido Schneider. “Breather solutions for a semi- linear Klein-Gordon equation on a periodic metric graph”. In:J. Math. Anal. Appl.528.2 (2023), Paper No. 127520, 31.issn: 0022-247X,1096-0813.doi:10.1016/j.jmaa.2023. 127520

work page doi:10.1016/j.jmaa.2023 2023

[14] [15]

Water waves, nonlinear Schrödinger equations and their solutions

D. H. Peregrine. “Water waves, nonlinear Schrödinger equations and their solutions”. In: The Journal of the Australian Mathematical Society. Series B. Applied Mathematics25.1 (1983), pp. 16–43.doi:10.1017/S0334270000003891

work page doi:10.1017/s0334270000003891 1983

[15] [16]

Michael Plum and Wolfgang Reichel.Breathers and rogue waves for semilinear curl-curl wave equations. Tech. rep. 73. Karlsruher Institut für Technologie (KIT), 2022. 22 pp. doi:10.5445/IR/1000153722

work page doi:10.5445/ir/1000153722 2022

[16] [17]

CRC 1173 Preprint 2025/40

Guido Schneider and Nils Thorin.Approximate Peregrine solitons in dispersive nonlinear wave equations. CRC 1173 Preprint 2025/40. Karlsruhe Institute of Technology, July 2025.doi:10.5445/IR/1000183475.url:https://www.waves.kit.edu/downloads/ CRC1173_Preprint_2025-40.pdf

work page doi:10.5445/ir/1000183475.url:https://www.waves.kit.edu/downloads/ 2025

[17] [18]

The method of Nehari manifold

Andrzej Szulkin and Tobias Weth. “The method of Nehari manifold”. In:Handbook of nonconvex analysis and applications. Int. Press, Somerville, MA, 2010, pp. 597–632.isbn: 978-1-57146-200-8

2010

[18] [19]

Theory of electron spin resonance in scanning tunneling mi- croscopy

C. B. Ward et al. “Rogue waves and periodic solutions of a nonlocal nonlinear Schrödinger model”. In:Phys. Rev. Res.2 (1 Mar. 2020), p. 013351.doi:10.1103/PhysRevResearch. 2.013351.url:https://link.aps.org/doi/10.1103/PhysRevResearch.2.013351

work page doi:10.1103/physrevresearch 2020

[19] [20]

Evaluating the robustness of rogue waves under perturbations

C.B. Ward, P.G. Kevrekidis, and N. Whitaker. “Evaluating the robustness of rogue waves under perturbations”. In:Physics Letters A383.22 (2019), pp. 2584–2588.issn: 0375- 9601.doi:https : / / doi . org / 10 . 1016 / j . physleta . 2019 . 05 . 030.url:https : //www.sciencedirect.com/science/article/pii/S0375960119304554

2019

[20] [21]

Michel Willem.Minimax theorems. Vol. 24. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 1996, pp. x+162.isbn: 0-8176-3913-6.doi:10.1007/978-1-4612-4146-1

work page doi:10.1007/978-1-4612-4146-1 1996

[21] [22]

Bo Yang and Jianke Yang.Rogue Waves in Integrable Systems /. 1st ed. 2024. Cham : Springer International Publishing, 2024.url:https://doi.org/10.1007/978-3-031- 66793-0

work page doi:10.1007/978-3-031- 2024