Characteristics of the Soliton Molecule and Lump Solution in the 2 + 1 -Dimensional Higher-Order Boussinesq Equation

The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the 2 + 1 -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2 + 1 -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual 2 + 1 -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.

Transformation of first mode breather of internal waves above the bottom step in a three-layer fluid

In the present study we consider propagation of a localized internal perturbation in the form of an oscillating wave packet (breather) of the first mode in a three-layer fluid with an uneven bottom shaped as a smoothed step. The study is carried out by methods of numerical simulation within a fully nonlinear two-dimensional (vertical plane) set of NavierStokes equations. A set of calculations was carried out for different widths and heights of the bottom step. Inhomogeneity of the medium leads to transformation of the internal wave field with the formation of weak reflected waves and one or two first-mode breathers passed to the shallow zone. By analyzing linear stability in terms of Richardson and Froude numbers, it was revealed that potentially unstable regions arise at the smallest values of the step width. An amplitude and energy analysis of secondary reflected nonlinear waves was performed. The vertical mode composition of the fully nonlinear wave field is analyzed. It is shown that the first mode makes the largest contribution to the vertical structure of the full-nonlinear packet, though the fourth, second and the third modes also contribute noticeably.

Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution

We study the occurrence and dynamics of rogue waves in three-dimensional deep water using phase-resolved numerical simulations based on a high-order spectral (HOS) method. We obtain a large ensemble of nonlinear wave-field simulations ($M= 3$ in HOS method), initialized by spectral parameters over a broad range, from which nonlinear wave statistics and rogue wave occurrence are investigated. The HOS results are compared to those from the broad-band modified nonlinear Schrödinger (BMNLS) equations. Our results show that for (initially) narrow-band and narrow directional spreading wave fields, modulational instability develops, resulting in non-Gaussian statistics and a probability of rogue wave occurrence that is an order of magnitude higher than linear theory prediction. For longer times, the evolution becomes quasi-stationary with non-Gaussian statistics, a result not predicted by the BMNLS equations (without consideration of dissipation). When waves spread broadly in frequency and direction, the modulational instability effect is reduced, and the statistics and rogue wave probability are qualitatively similar to those from linear theory. To account for the effects of directional spreading on modulational instability, we propose a new modified Benjamin–Feir index for effectively predicting rogue wave occurrence in directional seas. For short-crested seas, the probability of rogue waves based on number frequency is imprecise and problematic. We introduce an area-based probability, which is well defined and convergent for all directional spreading. Based on a large catalogue of simulated rogue wave events, we analyse their geometry using proper orthogonal decomposition (POD). We find that rogue wave profiles containing a single wave can generally be described by a small number of POD modes.