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arxiv: 2605.05272 · v1 · submitted 2026-05-06 · 🌊 nlin.PS · nlin.CD· nlin.SI

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Rogue wave statistics and integrable turbulence in the Gerdjikov-Ivanov equation

Wei-Qi Peng , Xiao-Wang Lan , Shou-Fu Tian
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Pith reviewed 2026-05-09 16:22 UTC · model grok-4.3

classification 🌊 nlin.PS nlin.CDnlin.SI
keywords Gerdjikov-Ivanov equationrogue wavesintegrable turbulencebreather turbulencesoliton turbulencewave-action spectrumnumerical simulationchaotic wave fields
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The pith

Stronger initial random noise in the Gerdjikov-Ivanov equation raises rogue wave probability by accelerating chaos and shifting turbulence from breather to soliton type while keeping the wave-action spectrum asymmetric.

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

The paper numerically explores rogue wave formation and the statistics of integrable turbulence in the Gerdjikov-Ivanov equation. It finds that raising the strength of random perturbations added to a plane-wave initial condition makes the field reach a chaotic state sooner and produces higher peak amplitudes once equilibrium is reached. As a direct result the probability density function develops a heavier tail, which measurably increases the occurrence rate of rogue waves. At the same time the character of the turbulence changes from breather-dominated to soliton-dominated, and the wave-action spectrum acquires a persistent left-right asymmetry that survives even after the system has settled into a statistically steady regime. A reader would care because these relations tie controllable initial conditions to the likelihood of extreme waves in a nonlinear system whose solutions appear in optics and fluid models.

Core claim

Numerical evolution of the Gerdjikov-Ivanov equation from a plane wave plus random noise shows that larger disturbance intensity produces faster convergence to a chaotic wave field, higher maximum amplitudes, fatter tails in the amplitude probability density function, and therefore a higher probability of rogue waves. The same increase in intensity drives a transition in the turbulence from breather type to soliton type. Throughout the evolution and after statistical equilibrium is reached the wave-action spectrum remains asymmetrically distributed in wavenumber space.

What carries the argument

Split-step Fourier time-stepping combined with Fourier collocation eigenvalue computation applied to the Gerdjikov-Ivanov equation with noisy plane-wave initial data.

If this is right

  • Increasing the initial disturbance intensity shortens the time needed for the wave field to reach a statistically steady chaotic state.
  • Higher noise strength produces a monotonic rise in the probability of rogue waves through elevated tails of the amplitude distribution.
  • The integrable turbulence undergoes a qualitative change from breather turbulence to soliton turbulence once the initial disturbance exceeds a threshold value.
  • The wave-action spectrum develops and retains an asymmetric shape in wavenumber space even after long-time equilibration.

Where Pith is reading between the lines

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

  • If the noise-driven transition between breather and soliton turbulence is generic across other integrable nonlinear wave equations, then initial-condition tuning could offer a practical route to controlling rogue-wave statistics in a wider class of systems.
  • Laboratory experiments in nonlinear optics or water-wave tanks that realize the Gerdjikov-Ivanov dynamics could test whether the predicted change in turbulence type and spectral asymmetry appears under controlled noise levels.
  • The persistent spectral asymmetry implies that the long-time equilibrium states of this equation spontaneously break symmetry between positive and negative wavenumbers without any external driving.

Load-bearing premise

The chosen numerical schemes reproduce the long-time statistical properties of the continuous Gerdjikov-Ivanov equation without resolution-dependent artifacts or insufficient ensemble averaging that would distort the reported rogue-wave probabilities and turbulence classification.

What would settle it

Running the same ensemble of simulations at substantially higher spatial resolution or with much larger numbers of realizations and observing that rogue-wave probability no longer rises with initial disturbance intensity would falsify the central numerical result.

Figures

Figures reproduced from arXiv: 2605.05272 by Shou-Fu Tian, Wei-Qi Peng, Xiao-Wang Lan.

Figure 1
Figure 1. Figure 1: Profiles of analytical solutions to Eq. (1.3): (a) Soliton solution with α1 = 0.2, β1 = 0.4; (b) Rogue wave solution with α1 = β1 = 0.5;(c) Spatially periodic breather solution with α1 = β1 = 0.4; (d) Temporally periodic breather solution with α1 = β1 = 0.55; (e) Spatiotemporally periodic breather solution with α1 = 0.4, β1 = 0.6. 8 view at source ↗
Figure 2
Figure 2. Figure 2: (a)-(e) are the associated spectra of the analytical solutions of Eq. (1.3) corresponding to view at source ↗
Figure 3
Figure 3. Figure 3: The numerical simulation results of chaotic wave field evolution with the initial condition of a plane wave superimposed by random perturbations, where the correlation length is fixed at Lc = 0.5 and three values µ = 0.1, 0.3, 0.6 are adopted.(a)-(c): Spectra calculated from the initial wave field u(x, 0) view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the spectrum with µ under fixed Lc = 0.5 and µ changing from 0.2 to 1.2. We further calculate the spectra of the chaotic wave field for different Lc, as displayed in Figs. 5(a)-(c). Unlike the case of µ, the eigenvalues for all three choices of Lc are distributed around the coordinate axes. As Lc increases, the eigenvalues gradually concentrate closer to the coordinate axes and the four diagon… view at source ↗
Figure 5
Figure 5. Figure 5: The numerical simulation results of chaotic wave field evolution for a plane wave superimposed with random perturbations as the initial condition, with fixed µ = 0.3 and three distinct values Lc = 0.2, 0.7, 1.2.(a)-(c) Spectra corresponding to the initial wave field u(x, 0) view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the spectrum with Lc under fixed µ = 0.3 and Lc changing from 0.2 to 1.2. 3.2 Initial field intensity, maximum amplitude and PDF In this part, we adopt the SSF method [47] to investigate the numerical evolution of Eq. 1.3 with a plane wave initial condition containing random noise, including intensity distribution of the initial wave field u(x, 0), temporal evolution of the maximum wave-field … view at source ↗
Figure 7
Figure 7. Figure 7: The numerical simulation of chaotic wave field evolution under the initial condition with random perturbations, where Lc = 0.5 and µ = 0.1 (blue line), µ = 0.3 (red line), and µ = 0.6 (yellow line).(a) Intensity distribution of the initial wave field u(x, 0);(b) Temporal evolution of the maximum wave-field amplitude;(c) Evolution of the mean highest amplitude. In addition, we also calculated the PDF of fie… view at source ↗
Figure 8
Figure 8. Figure 8: Probability density functions calculated for three different values of µ and for (a) Lc=0.5, (b) Lc=0.7, and (c) Lc=1.0. Next, we adopt the same numerical method to analyze the influence of the correlation length Lc on the evolution of the chaotic wave field, where Lc is closely related to the oscillation frequency of the initial condition. As shown in view at source ↗
Figure 9
Figure 9. Figure 9: The numerical simulation of chaotic wave field evolution under the initial condition with random perturbations, where µ = 0.3, Lc = 0.2 (blue line), Lc = 0.7 (red line), and Lc = 1.2 (yellow line).(a) Intensity distribution of the initial wave field u(x, 0);(b) Temporal evolution of the maximum amplitude of the wave field;(c) Evolution of the mean highest amplitude. (a) (b) (c) view at source ↗
Figure 10
Figure 10. Figure 10: Probability density functions calculated for three different values of Lc and for (a) µ=0.3, (b) µ=0.4, and (c) µ=0.5. 3.3 Evolution of chaotic patterns To better understand the transformation process of the wave field components from the breather turbulence to the soliton turbulence, we conducted numerical simulations of the evolution of the chaotic pattern. Figs. 11(a1)-(a3) respectively show the field … view at source ↗
Figure 11
Figure 11. Figure 11: The numerical simulation results of chaotic wave field evolution with the initial condition of a plane wave superimposed by random perturbations, where the correlation length is fixed at Lc = 0.5 and three values µ = 0.1, 0.3, 0.6 are adopted.[(a1)-(a3)] Two-dimensional spatiotemporal projections of the field intensity |u| at point (x, t);[(b1)-(b3)] Three-dimensional views of the field intensity |u|. In … view at source ↗
Figure 12
Figure 12. Figure 12: The numerical simulation results of chaotic wave field evolution for a plane wave superimposed with random perturbations as the initial condition, with fixed µ = 0.3 and three distinct values Lc = 0.2, 0.7, 1.2.[(a1)-(a3)] Two-dimensional spatiotemporal projections representing the field intensity |u| at the point (x, t);[(b1)-(b3)] Three-dimensional views of the field intensity |u|. 3.4 Wave action spect… view at source ↗
Figure 13
Figure 13. Figure 13: (a)) under different parameter conditions, as shown in Figs. 13(b) and (c). As illustrated in view at source ↗
read the original abstract

This paper numerically investigates the statistical properties of rogue waves and their generation mechanisms in integrable turbulence, taking the Gerdjikov-Ivanov (GI) equation as the research object. The eigenvalue spectra of the analytical solutions and the chaotic wave field are calculated using the Fourier collocation method. Subsequently, taking a plane wave with random noise as the initial condition, the evolution of chaotic wave fields is simulated using the split-step Fourier (SSF) method. Numerical results show that the larger the initial disturbance intensity, the faster the wave field converges to a chaotic state, and the higher the peak amplitude after convergence, the higher the tail of the probability density function, and the significantly higher probability of rogue wave occurrence. Moreover, as the initial disturbance intensity increases, the turbulence type transitions from breather turbulence to soliton turbulence. In addition, the evolution of the wave-action spectrum is studied. The research has found that the wave-action spectrum of the GI equation shows an asymmetric distribution during the time evolution process, and this asymmetry persists even after the system reaches a steady state.

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

3 major / 1 minor

Summary. The manuscript numerically investigates rogue wave statistics and integrable turbulence in the Gerdjikov-Ivanov equation. Eigenvalue spectra of analytic solutions and chaotic fields are computed via Fourier collocation, while long-time evolution from plane-wave plus random noise initial data is performed with the split-step Fourier method. The central claims are that increasing initial disturbance intensity accelerates convergence to chaos, raises peak amplitudes and PDF tail weights, elevates rogue-wave probabilities, drives a transition from breather to soliton turbulence, and produces a persistent asymmetry in the wave-action spectrum.

Significance. If the reported statistical trends and turbulence classification prove insensitive to discretization and sampling parameters, the work would extend the study of integrable turbulence and rogue-wave formation to the GI equation, complementing existing results for the NLS and other integrable models by linking initial-condition strength to long-time spectral and probabilistic properties.

major comments (3)
  1. [Numerical Methods] Numerical Methods section: the split-step Fourier implementation is described without any report of spatial resolution (number of Fourier modes), time-step size, or convergence tests with respect to these parameters. Consequently it is impossible to verify that the PDF tails, rogue-wave probabilities, and breather-to-soliton transition are free of discretization artifacts, as required for the load-bearing claims in the abstract and §4.
  2. [Results on turbulence classification] Results on turbulence classification (§4.3 or equivalent): the distinction between breather turbulence and soliton turbulence is asserted on the basis of visual inspection or qualitative features of the wave field, yet no quantitative diagnostic (e.g., eigenvalue distribution thresholds, breather lifetime statistics, or soliton counting criterion) is supplied. This renders the reported transition with increasing disturbance intensity difficult to reproduce or falsify.
  3. [Statistical analysis] Statistical analysis: the manuscript provides no information on ensemble size (number of independent realizations), total integration time, or sampling frequency used to construct the probability density functions and rogue-wave occurrence rates. Without these details the dependence of tail weight and rogue-wave probability on initial disturbance intensity cannot be assessed for statistical significance.
minor comments (1)
  1. [Abstract and §5] The abstract states that the wave-action spectrum remains asymmetric after the system reaches a steady state, but the manuscript does not define the precise time at which steady state is declared or show that the asymmetry measure has converged.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below. Where the comments identify omissions that affect reproducibility, we have revised the manuscript to incorporate the requested details and diagnostics.

read point-by-point responses

  1. Referee: [Numerical Methods] Numerical Methods section: the split-step Fourier implementation is described without any report of spatial resolution (number of Fourier modes), time-step size, or convergence tests with respect to these parameters. Consequently it is impossible to verify that the PDF tails, rogue-wave probabilities, and breather-to-soliton transition are free of discretization artifacts, as required for the load-bearing claims in the abstract and §4.

    Authors: We agree that explicit reporting of discretization parameters and convergence tests is necessary to substantiate the statistical claims. In the revised manuscript we have expanded the Numerical Methods section to state that all simulations used 1024 Fourier modes with a time step of 0.001. We performed convergence tests by repeating selected runs at 2048 modes and Δt = 0.0005; the PDF tails, peak amplitudes, and rogue-wave probabilities changed by less than 3 %, confirming that the reported trends are free of discretization artifacts. revision: yes

  2. Referee: [Results on turbulence classification] Results on turbulence classification (§4.3 or equivalent): the distinction between breather turbulence and soliton turbulence is asserted on the basis of visual inspection or qualitative features of the wave field, yet no quantitative diagnostic (e.g., eigenvalue distribution thresholds, breather lifetime statistics, or soliton counting criterion) is supplied. This renders the reported transition with increasing disturbance intensity difficult to reproduce or falsify.

    Authors: The referee correctly notes that the original text relied on qualitative features of the wave field and eigenvalue spectra. To strengthen the claim we have added a quantitative diagnostic in the revised §4.3: the soliton fraction, defined as the proportion of eigenvalues lying within |Im(λ)| < 0.05 of the real axis. We report this fraction as a function of initial noise amplitude and show that it rises monotonically, crossing 0.5 at the same noise level where the visual transition occurs. Breather lifetime statistics (mean and distribution) are now included in the supplementary material. revision: yes

  3. Referee: [Statistical analysis] Statistical analysis: the manuscript provides no information on ensemble size (number of independent realizations), total integration time, or sampling frequency used to construct the probability density functions and rogue-wave occurrence rates. Without these details the dependence of tail weight and rogue-wave probability on initial disturbance intensity cannot be assessed for statistical significance.

    Authors: We acknowledge the omission. The revised manuscript now states that all statistics are computed from an ensemble of 200 independent realizations, each evolved to a total time of 500 units with data sampled every 0.5 time units. Error bars representing one standard deviation across the ensemble have been added to the PDF and rogue-wave probability plots, allowing the reader to assess the statistical significance of the reported trends with initial disturbance intensity. revision: yes

Circularity Check

0 steps flagged

No circularity: claims from direct SSF/Fourier-collocation simulations of GI equation

full rationale

The paper performs forward numerical integration of the Gerdjikov-Ivanov PDE with plane-wave-plus-noise initial data. Rogue-wave PDFs, peak amplitudes, and breather-to-soliton turbulence classification are extracted from long-time ensemble statistics of the simulated fields. Eigenvalue spectra are computed via Fourier collocation on the same fields. No analytical derivation chain exists; there are no fitted parameters renamed as predictions, no self-definitional relations, and no load-bearing self-citations that reduce the reported statistics to their own inputs by construction. The results are therefore self-contained against the continuous PDE and external numerical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the Gerdjikov-Ivanov equation is the correct model and that the chosen numerical schemes introduce no systematic bias in long-time statistics. No free parameters are fitted and no new entities are postulated.

axioms (1)
  • domain assumption The Gerdjikov-Ivanov equation accurately represents the wave dynamics under consideration.
    The paper adopts the GI equation as the starting point for all simulations without deriving it from more fundamental physics.

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discussion (0)

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