Evolution of perturbed long nonlinear plane, ring and hybrid surface waves

TL;DR

This work tackles the two-dimensional evolution of perturbed long nonlinear surface waves within the 2D Boussinesq–Peregrine framework and introduces the $KdV\theta$ amplitude equation to capture intermediate 2D dynamics. By deriving and linking the axisymmetric $cKdV$, the $KdV\theta$ equation, and KPII, the authors construct a unified reduced-model perspective that is validated against full 2D simulations for plane, ring, and hybrid wave configurations. Key findings show outward-propagating ring and hybrid waves to be stable, while inward-propagating configurations can exhibit rogue-wave–like phenomena, including lump formation and an eventual X-type KPII pattern, highlighting possible rogue-wave mechanisms. The work provides analytical mappings (via inverse scattering) and numerical evidence for the applicability of reduced models to 2D wave evolution, with clear paths for extensions to stratified fluids and shear flows.

Abstract

The two-dimensional evolution of perturbed long weakly-nonlinear surface plane, ring, and hybrid waves, consisting, to leading order, of a part of a ring and two tangent plane waves, is modelled numerically within the scope of the 2D Boussinesq-Peregrine system. Numerical runs are initiated and interpreted using the reduced 2+1-dimensional cKdV-type and KPII equations. The cKdV-type equation leads to two different models, the KdV$θ$ and cKdV equations, depending on whether we use the general or singular (i.e. the envelope of the general) solution of the associated nonlinear first-order differential equation. The KdV$θ$ equation is also derived directly from the 2D Boussinesq-Peregrine system and used to analytically describe the intermediate 2D asymptotics of line solitons subject to sufficiently long transverse perturbations of finite strength, while the cKdV equation is used to initiate outward- and inward-propagating ring waves with localised and periodic perturbations. Both of these equations, together with the KPII equation, are used to model the evolution of hybrid waves, where we show, in particular, that large localised waves (lumps) can appear as transient (emerging and then disappearing) states in the evolution of inward-propagating waves, contributing to the possible mechanisms for the generation of rogue waves. Detailed comparisons are made between the key features of the non-stationary two-dimensional modelling and relevant predictions of the reduced equations.

Figures (18)

• Figure 1: Hybrid wave generated in the Strait of Gibraltar (Terra MODIS, 22 May 2017, 11:30 UTC. NASA ESDIS Worldview, https://user.eumetsat.int/resources/case-studies/internal-waves-in-the-eastern-strait-of-gibraltar).
• Figure 2: Wavefront and wave vector of a plane wave propagating at an angle $\varphi$ to the direction of the shear flow.
• Figure 3: $(a)$ The general solution (\ref{['gs']}) for $a = 0.7$ (blue), $a=0.8$ (dashed blue) and $a=0.9$ (dashed dot blue) and its envelope (the singular solution, solid red) all for $u_0(z) = \gamma z$, $\gamma = 0.2$. $(b)$ The wavefronts of the corresponding general and singular solutions from $(a)$ described by $rk(\theta) = 5$.
• Figure 4: Plots of the numerical solution of both the KdV$\theta$ equation and 2D Boussinesq--Peregrine system for a perturbed line soliton with the parameters $\epsilon = 0.05$, $\tilde{v} = 0.5$, $\xi_0 = 180$, $\alpha = 2$, $\beta = 5$ between $T_0 = 1$ and $T = 6$. The dark grey (dash-dotted) and the light grey (dashed) lines illustrate theoretical (IST) predictions for the perturbed regions of the main and secondary solitons, respectively.
• Figure 5: Convergence of the solution of the KdV$\theta$ equation to the solution of the 2D Boussinesq--Peregrine system for different lengths of perturbation and $\epsilon$, where $||d||_\infty$ is the $L^\infty$ norm of the difference between solutions of the 2D Boussinesq--Peregrine system and KdV$\theta$ equation, and $T \in [1,2]$, $\tilde{v} = 0.5$, $\xi_0 = 200$, and $\alpha = 0.5$.