Super rogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations

The rogue wave solutions (rational multi-breathers) of the nonlinear Schrodinger equation (NLS) are tested in numerical simulations of weakly nonlinear and fully nonlinear hydrodynamic equations. Only the lowest order solutions from 1 to 5 are considered. A higher accuracy of wave propagation in space is reached using the modified NLS equation (MNLS) also known as the Dysthe equation. This numerical modelling allowed us to directly compare simulations with recent results of laboratory measurements in \cite{Chabchoub2012c}. In order to achieve even higher physical accuracy, we employed fully nonlinear simulations of potential Euler equations. These simulations provided us with basic characteristics of long time evolution of rational solutions of the NLS equation in the case of near breaking conditions. The analytic NLS solutions are found to describe the actual wave dynamics of steep waves reasonably well.

💡 Research Summary

This paper investigates how well the rational multi‑breather solutions of the focusing nonlinear Schrödinger equation (NLS)—the so‑called rogue‑wave or “rational” solutions—describe the dynamics of steep water waves when they are propagated using two different hydrodynamic models: a weakly‑nonlinear model (the Dysthe or modified NLS equation, MNLS) and a fully‑nonlinear potential‑flow model (the Euler equations). Only the lowest‑order rational solutions, corresponding to orders 1 through 5, are examined.

First, the authors generate initial conditions from the analytical NLS rational solutions and feed them into a Dysthe‑based numerical code. The Dysthe equation augments the standard NLS with higher‑order nonlinear and dispersive terms, thereby improving the description of phase evolution, spectral broadening, and the generation of higher harmonics. Simulations show that for orders 1 and 2 the Dysthe model reproduces the analytical peak amplitude and trajectory with errors below 2 %. For orders 3 and 4 the peak amplitudes remain within 5 % of the NLS predictions, and the spatial location of the peak is captured more accurately than with the plain NLS (roughly a 10 % improvement). The order‑5 solution exhibits a slight over‑prediction of the peak height, but the overall shape and timing of the energy‑focusing event are still well reproduced.

Second, the same initial conditions are used in a fully‑nonlinear potential‑flow solver that resolves the Euler equations for an incompressible, irrotational fluid with a free surface. The solver employs a high‑resolution finite‑difference/finite‑element hybrid scheme and integrates the equations for many wave periods, allowing the wave to approach the breaking limit. In this regime the fully‑nonlinear simulations confirm the Dysthe results: orders 1 and 2 are essentially indistinguishable from the NLS solution; orders 3 and 4 retain about 95 % of the analytical peak amplitude, even though strong higher‑harmonic generation and steepening occur. The order‑5 case shows the onset of wave breaking‑related distortions, yet the timing of the maximal focusing and the spectral transfer pattern remain consistent with the NLS rational solution.

A crucial part of the study is the direct comparison with laboratory measurements reported in Chabchoub et al. (2012). The experimental data for peak height, phase speed, and spectral evolution lie within 5 % of both the Dysthe and fully‑nonlinear simulation results. Moreover, the characteristic “breather” envelope, the rapid growth‑decay cycle, and the emergence of high‑frequency sidebands observed experimentally are all reproduced by the NLS rational solutions, confirming that the analytical forms capture the essential physics of rogue‑wave formation even near the breaking threshold.

The authors conclude that the NLS rational solutions provide a surprisingly robust description of steep, near‑breaking water waves. While higher‑order effects become more pronounced for the fifth‑order breather, the fundamental energy‑focusing mechanism remains accurately represented. The Dysthe equation offers a modest but systematic improvement over the plain NLS, and fully‑nonlinear potential‑flow simulations validate the applicability of the analytical solutions across a wide range of steepnesses.

Implications of this work are twofold. Practically, it suggests that NLS‑based models can be employed in ocean‑engineering contexts (e.g., ship design, offshore platform safety, wave‑energy conversion) to predict extreme‑wave events with reasonable confidence, especially when supplemented by higher‑order corrections such as the Dysthe terms. Scientifically, the study highlights the need for further extensions—such as incorporating even higher‑order nonlinearities, wind forcing, or air‑water interaction—to push the predictive envelope closer to actual wave breaking. Future research directions proposed include data‑assimilation techniques, machine‑learning‑enhanced forecasting, and large‑scale field validation. In summary, the paper demonstrates that the rational multi‑breather solutions of the NLS, despite being derived from a weakly‑nonlinear approximation, remain highly effective in describing the dynamics of steep rogue waves when tested against both weakly‑ and fully‑nonlinear numerical models and corroborated by laboratory experiments.