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arxiv: 2605.26961 · v1 · pith:AN3P6DCKnew · submitted 2026-05-26 · 🌊 nlin.PS · math-ph· math.MP

x-periodic Quasi One Dimensional Anomalous (Rogue) Waves in Multidimensional Nonlinear Schr\"odinger Equations: Fission, Fusion, and Recurrence

Francesco Coppini , Paolo Maria Santini This is my paper

Pith reviewed 2026-06-29 14:30 UTC · model grok-4.3

classification 🌊 nlin.PS math-phmath.MP
keywords rogue wavesanomalous wavesmodulation instabilitynonlinear Schrödinger equationsquasi-one-dimensional regimefission and fusionrecurrence dynamicsfinite-gap perturbation
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The pith

Recurrence of x-periodic rogue waves in multidimensional NLS equations shows O(1) model-specific differences and increasingly complex fission-fusion after a universal first stage.

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

The paper studies the recurrence of x-periodic anomalous waves in the quasi-one-dimensional regime of several multidimensional nonlinear Schrödinger equations. It establishes that the initial nonlinear stage of modulation instability remains essentially universal across models and is captured by deformations of the Akhmediev breather, yet the subsequent recurrence dynamics differ by order-one amounts between models. Successive stages display progressively richer combinations of fission and fusion, and the authors derive an analytic description of these recurrences by treating the equations as multidimensional perturbations of the integrable NLS and applying finite-gap perturbation theory, with results matching numerical simulations.

Core claim

Although the first nonlinear stage of MI is essentially universal for all MNLS equations, the recurrence dynamics exhibit significant O(1) differences among different models. Moreover, successive nonlinear stages generally display increasingly complex combinations of fission and fusion processes, leading to progressively richer dynamical choreographies. Since MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation, the recently developed finite gap perturbation theory of NLS AWs gives an analytic and quantitative description of the recurrence of Q1D AWs, in excellent agreement with numerical simulations.

What carries the argument

Finite-gap perturbation theory of NLS anomalous waves, used to describe multidimensional perturbations in the Q1D regime and track recurrence via fission and fusion.

If this is right

  • Recurrence dynamics exhibit O(1) differences among different MNLS models after the shared initial stage.
  • Successive nonlinear stages display increasingly complex fission and fusion choreographies.
  • The finite-gap perturbation theory supplies an analytic description that matches numerical simulations of the recurrence.
  • The described processes are plausible to observe in water waves, nonlinear optics, plasma physics and Bose-Einstein condensates.

Where Pith is reading between the lines

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

  • Long-term rogue-wave behavior may be more sensitive to the precise form of the MNLS model than the onset of instability itself.
  • The same perturbation approach could be applied to other regimes or higher-dimensional perturbations to predict similar choreography changes.
  • Experiments could distinguish between candidate MNLS models by measuring the detailed recurrence patterns rather than only the first instability.

Load-bearing premise

MNLS equations in the Q1D regime can be treated as multidimensional perturbations of the integrable NLS equation.

What would settle it

Numerical evolution of recurrence in a specific MNLS model that deviates by more than O(1) from the finite-gap perturbation predictions while the first MI stage remains universal.

Figures

Figures reproduced from arXiv: 2605.26961 by Francesco Coppini, Paolo Maria Santini.

Figure 1
Figure 1. Figure 1: The density plot (left) and the 3D plot (right) of the NLS numerical [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The infinitely many spines near the real axis of the left picture are [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The schematic representation of the first nonlinear stages of MI for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The two-step recurrence scheme describing the AW recurrence of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A local description of the process of “growth from the background [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A local description of the process of “fusion + decay to the back [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Three snapshots of |u1(x, Y, t)| 2 describing locally the fission pro￾cess in the Y direction. From left to right, t = −0.1: the AW growth, t = 0 the AW fission, t = 0.1: the separation of the fission products. Here Lx = 0 ⇒ ϕ = 1.02, t¨1(0) = 2, x˙ 1(0) = ¨x1(0) = 0. Changing τ → −τ and fiss → fus, equations (79),(80) give the local description of fusion. Since AW fission is not restricted to the Q1D regi… view at source ↗
Figure 8
Figure 8. Figure 8: The first NLSMI for both HNLS and ENLS equations. The graph [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The first NLSMI for both HNLS and ENLS equations. Six snap [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The second NLSMI for the HNLS equation. The upper picture [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The second NLSMI for the HNLS equation. Eight snapshots of [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the second NLSMI for the ENLS and HNLS equa [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Numerical evolution according to the HNLS equation of the real [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Upper figure: the plot of t1(Y ), together with three horizontal dashed lines at t = 2.8, tf iss = 3.005, 3.5 in the (Y, t) plane; lower figure: the parabolic plot of x1(Y ) in the (Y, x) plane. The initial data are c0+ = 0.3 + i 0.2 and c0− = i 0.5, with Lx = 6 and ϵ = 0.01, δ = 0.001. It follows that the growing AW u1(x, Y, t) is exponentially localized over the background in the Y direction, undergoing… view at source ↗
Figure 15
Figure 15. Figure 15: The first NLSMI. 3 snapshots of |u(x, Y, t)| at the time corre￾sponding to the three horizontal dashed lines of [PITH_FULL_IMAGE:figures/full_fig_p033_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The second NLSMI for both equations. The upper left pic [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The second NLSMI for the HNLS equation. Three snapshots of [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The second NLSMI for the ENLS equation. Height snapshots [PITH_FULL_IMAGE:figures/full_fig_p036_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The uniform distance between the numerical output and the [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The uniform distance between the numerical output and the the [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
read the original abstract

In a recent work we studied the first nonlinear stage of modulation instability (NLSMI) of x-periodic anomalous (rogue, freak, extreme) waves (AWs) of physically relevant multidimensional (generalizations of the focusing) nonlinear Schr\"odinger (MNLS) equation, like the non integrable elliptic and hyperbolic nonlinear Schr\"odinger (NLS) equations in d+1 dimensions, d = 2, 3, in the quasi one dimensional (Q1D) regime in which the wavelength in the direction of propagation x is small with respect to the wavelengths in the transversal directions. We showed that, at leading order, the first NLSMI is universal, independent of the particular MNLS model, and described by suitable adiabatic deformations of the quasi-homoclinic Akhmediev breather solution of NLS, in excellent agreement with numerical simulations. In the present work we focus on the recurrence of x-periodic AWs in the Q1D regime. We show that, although the first nonlinear stage of MI is essentially universal for all MNLS equations, the recurrence dynamics exhibit significant O(1) differences among different models. Moreover, successive nonlinear stages generally display increasingly complex combinations of fission and fusion processes, leading to progressively richer dynamical choreographies. Since MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation, we use the recently developed finite gap perturbation theory of NLS AWs to give an analytic and quantitative description of the recurrence of Q1D AWs, in excellent agreement with numerical simulations. Due to the physical relevance of the MNLS equations considered in this work, and due to the universality of the processes discussed in this paper, it is plausible that they be observable in many fields of physics, like water waves, nonlinear optics, plasma physics, Bose-Einstein condensates, etc . . .

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.

Referee Report

1 major / 1 minor

Summary. The manuscript studies recurrence of x-periodic anomalous (rogue) waves in multidimensional nonlinear Schrödinger equations (including non-integrable elliptic and hyperbolic cases) in the quasi-one-dimensional (Q1D) regime. It asserts that the first nonlinear stage of modulation instability remains essentially universal across models and is captured by adiabatic deformations of the Akhmediev breather, while recurrence exhibits significant O(1) model-dependent differences, with successive stages displaying increasingly complex fission/fusion processes. The authors apply recently developed finite-gap perturbation theory of NLS anomalous waves to furnish an analytic, quantitative description of these recurrence dynamics, claiming excellent agreement with numerical simulations.

Significance. If the finite-gap perturbation remains controlled over multiple recurrence stages, the work would supply a concrete analytic framework distinguishing universal from model-specific features of rogue-wave recurrence in physically relevant MNLS systems, with plausible observability in water waves, optics, plasmas, and BECs. The explicit demonstration of O(1) differences among models and the use of perturbation theory for quantitative recurrence predictions constitute the main potential contributions.

major comments (1)
  1. [Sections applying finite-gap perturbation theory to recurrence dynamics] The central claim that finite-gap perturbation theory yields a quantitative description of recurrence (including multi-stage fission/fusion choreographies) rests on the assumption that MNLS equations in the Q1D regime act as controlled perturbations of the integrable NLS. However, no a-priori estimate is given for the size of the transversal perturbation parameter or for the number of nonlinear stages over which the leading-order description remains valid. Each recurrence stage can amplify the perturbation, so the absence of such control directly affects the asserted excellent agreement beyond the first stage.
minor comments (1)
  1. [Abstract] The abstract states that the first NLSMI stage is 'essentially universal' while recurrence shows 'significant O(1) differences'; a brief clarification of the precise sense in which the perturbation remains small enough for the leading-order theory to capture O(1) model differences would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Sections applying finite-gap perturbation theory to recurrence dynamics] The central claim that finite-gap perturbation theory yields a quantitative description of recurrence (including multi-stage fission/fusion choreographies) rests on the assumption that MNLS equations in the Q1D regime act as controlled perturbations of the integrable NLS. However, no a-priori estimate is given for the size of the transversal perturbation parameter or for the number of nonlinear stages over which the leading-order description remains valid. Each recurrence stage can amplify the perturbation, so the absence of such control directly affects the asserted excellent agreement beyond the first stage.

    Authors: We agree that the manuscript provides no a priori estimates for the size of the transversal perturbation parameter or the number of nonlinear stages over which the leading-order finite-gap perturbation description remains valid. The Q1D MNLS equations are viewed as perturbations of the integrable NLS, but error accumulation across recurrence stages is not controlled rigorously. The quantitative agreement is demonstrated numerically for the specific models, parameters, and stages simulated in the figures. We will revise the manuscript to add an explicit discussion of this limitation, qualifying the applicability of the perturbation theory to the observed recurrence stages rather than claiming general control. revision: yes

Circularity Check

1 steps flagged

Recurrence description reduces to self-cited finite-gap perturbation theory of NLS AWs

specific steps
  1. self citation load bearing [Abstract]
    "Since MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation, we use the recently developed finite gap perturbation theory of NLS AWs to give an analytic and quantitative description of the recurrence of Q1D AWs, in excellent agreement with numerical simulations."

    The quantitative analytic description of recurrence (the paper's main result beyond the first stage) is obtained by direct application of the finite-gap theory developed in prior work. The step reduces the claimed description to that self-cited framework by construction; no independent derivation or a-priori error bound for multi-stage dynamics is supplied in the quoted text.

full rationale

The paper's central analytic claim for recurrence (including model-dependent O(1) differences and fission/fusion sequences) is obtained by viewing MNLS as perturbations of NLS and directly applying the 'recently developed finite gap perturbation theory of NLS AWs'. This theory is invoked as the source of the quantitative description without re-derivation or external verification in the present work. The abstract explicitly states the reduction: the Q1D regime 'can be viewed as multidimensional perturbations' and 'we use the recently developed finite gap perturbation theory' to describe recurrence. This matches self-citation load-bearing (pattern 3) because the load-bearing step for the claimed analytic results is the prior theory whose authors overlap. No independent bounds or machine-checked support for multi-stage validity is quoted. The first-stage universality is stated separately and does not rescue the recurrence claim. Score 6 reflects one load-bearing self-citation chain with partial independence in the numerical comparisons.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters or invented entities listed. The perturbation view of MNLS as multidimensional corrections to NLS is treated as a domain assumption.

axioms (1)
  • domain assumption MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation
    Stated explicitly in abstract as the basis for applying finite gap perturbation theory.

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