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

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$\theta$ 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$\theta$ 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.