Bright and dark rogue internal waves: The Gardner equation approach

Dynamical and statistical features of soliton interactions in the focusing Gardner equation

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.

Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

Internal waves in a stratified fluid with a constant buoyancy frequency were studied, with special attention given to rogue modes, extreme waves, dynamical evolution, and Fermi–Pasta–Ulam–Tsingou type recurrence phenomena. Rogue waves for triads in a general physical setting have recently been derived analytically, but the implications in a fluid mechanics context have not yet been fully assessed. Numerical simulations were conducted for cases of coupled triads where the common member is a daughter wave mode. In sharp contrast with previous studies, rogue modes instead of plane waves were used as the initial condition. Furthermore, spatial dependence was incorporated. Rogue or extreme waves in one set of triads provided a possible mechanism for significant energy transfer among modes of the internal wave spectrum, in addition to the other known theories, e.g., weak turbulence. Remarkably, Fermi–Pasta–Ulam–Tsingou recurrence types of growth and decay cycles arose, similar to those observed for surface gravity wave groups governed by the cubic nonlinear Schrödinger equation. These mechanisms will enhance our understanding of transport processes in oceans.