An idealized numerical model of one-dimensional shallow-water waves showed a dominant, or “champion,” soliton in each direction of propagation during the run that included weak damping at high wavenumbers. The modeled crest-height ratio, ηmax/Hs, was approximately 3, compared with the study’s reported crest-based rogue-wave criterion of ηmax/Hs greater than 1.25. The result was reported from a numerical trajectory and figure, without a distribution or confidence interval. It describes a pattern inside the model, not a measured wave in a physical shallow-water environment.
The study’s central question was whether random waves were involved in soliton intensification. In a mechanism comparison built around a 32-soliton ensemble, the researchers examined a soliton-only version and a version with an added bidirectional random-wave component. The soliton-only ensemble propagated essentially unchanged and did not form a champion. The random-wave-augmented case showed champion emergence and long-lived propagation. The comparison had no reported replication count or formal inferential analysis, so it does not provide a general effect size.
How the calculation was set up
The simulated system was the bidirectional Kaup–Boussinesq, or KB, wave model. The calculation added a weak dissipative perturbation at high wavenumbers, specified to break the equations’ underlying integrability. The wave field was then evolved numerically with a pseudospectral, fourth-order Runge–Kutta integration-factor method. That approach represents the field through spectral components and advances them through the model’s time evolution.
Solitons were defined through the direct scattering transform, also called the nonlinear Fourier transform. The analysis followed the bound states identified by that transform, along with their peak amplitudes, their number over time, the share of total system energy held by solitons, and changes during selected interactions. The wider set of calculations included bidirectional KB fields, a 32-soliton mechanism comparison, and sensitivity cases involving different dissipation settings and an unidirectional KdV field.
The conservative run supplied the baseline for the comparison. Without dissipation, the random field remained random and the discrete spectrum remained unchanged at the later observation, a pattern consistent with isospectral dynamics. In this context, that term means the model retained the same discrete spectral information as it evolved. The calculation therefore did not show preferential spectral growth that would single out a new dominant bound state in the conservative case.
A sequence of damping, growth and decline
The dissipative run began with a less dramatic pattern. It initially showed fewer bound states and lower peak amplitudes for those states, while both the random-wave component and the soliton component were dissipated. The early phase looked like a relaxation of the field rather than an immediate amplification. The later appearance of a champion came after this initial weakening within the modeled time evolution.
The intensification stage then culminated in a champion soliton in each propagation direction. The modeled value of ηmax/Hs reached approximately 3, above the reported crest-based threshold of greater than 1.25. That comparison places the simulated crest above the criterion used in the paper’s description, but it remains a classification generated from the model. It does not establish a physical rogue-wave incidence rate or show that the same crest behavior occurs in natural settings.
The dominant amplitudes did not rise without limit. During a quasi-steady stage, champion amplitudes were nearly constant when dissipative losses and interaction gains were nearly balanced. Later, champion intensities decreased, while the fraction of total energy held by solitons approached a plateau. The reported sequence was therefore early damping, subsequent intensification, approximate balance and eventual decline—not indefinite persistence of the champion structures.
The interaction pattern behind the proposed mechanism
The selected interaction examples showed a sharp difference between collisions in opposite and matching directions. Counter-propagating collisions were brief, lasting O(1 T_p), with virtually unchanged profiles and no significant energy transfer. A co-propagating pair remained together for longer than O(10 T_p). In that prolonged example, the dominant soliton gained energy while the subdominant soliton lost energy.
The paper presents the longer co-propagating encounters as a possible route for energy to move toward a dominant soliton. It also proposes that a random-wave background can help maintain weaker bound states, creating conditions in which the dissipative run departs from the conservative pattern. These are interpretations of the simulated trajectories and energy calculations. The comparison shows the association among the random-wave background, prolonged interactions and champion emergence, but it does not establish the same transfer in physical water.
The authors report qualitatively similar long-time behavior across distinct initial random-phase realizations. Slightly stronger dissipation appeared to accelerate champion formation in the reported tests. The outcome changed, however, when dissipation was stronger or acted across a larger scale: solitons could then be rapidly damped. The result suggests that the modeled behavior depends on how dissipation is configured, rather than identifying one setting that works uniformly across conditions.
Why the result remains provisional
There is a technical qualification around the soliton-identification method. The standard direct scattering transform assumes a problem on the real line, whereas the numerical solutions here are periodic. The authors therefore note that the transform may not be exact for these simulations, even though they regard it as effective for detecting bound states. That caveat matters because the number, trajectory and energy assigned to individual solitons depend on the method used to separate them from the full field.
The reported champion ratio came from a numerical trajectory rather than a distribution of repeated outcomes, and the study reports no formal statistical uncertainty, confidence intervals, p-values or effect-size distributions. The random-wave comparison likewise has no reported replication count or inferential analysis. The sensitivity results are qualitative, so they do not establish the dissipation thresholds at which champion formation accelerates or solitons are rapidly damped.
The evidence is limited to idealized, dimensionless one-dimensional KB and KdV numerical models on periodic domains. There were no physical shallow-water experiments or field observations, and the crest-based classification was not a direct observation of a measured rogue wave. The simulations also had no forcing, so the structures studied are distinct from gain-sustained dissipative solitons. The findings should therefore be read as a possible mechanism in a mathematical model, not as proof of a real-world hazard.
Further work would need to test whether the pattern survives in physical shallow-water experiments, more realistic viscous models, higher dimensions and broader parameter ranges. Systematic studies would also need to vary dissipation strength and scale, the random-wave spectrum, numerical resolution and independent initial conditions. Another open question is whether the reported crest-ratio behavior predicts measurable rogue-wave occurrence or hazard in natural settings.
Publication status
The manuscript is a preprint on arXiv, and its front matter says it is under consideration for publication in J. Fluid Mech. No acceptance decision is reported. The research was supported by the Simons Foundation through Award ID #651459, and the simulations used the Great Lakes HPC Cluster provided by Advanced Research Computing at the University of Michigan. The authors report no conflict of interest.
Paper data and sources
Original title: Dissipation-driven champion solitons in one-dimensional shallow-water waves
Authors: Ashleigh Simonis, Sergey Nazarenko, Jalal Shatah, Yulin Pan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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