Three dimensional hydroelastic solitary waves in shallow water #

Yanghan Meng, Zhan Wang

15:50 Monday in 2Q42.

Part of the Advances in water waves and free-surface flows session.

Abstract #

Solitary waves propagating on the surface of a three-dimensional ideal fluid bounded above by an elastic sheet are studied. Considered the complexity of the full Euler equations, a Benney-Luke-type equation is derived via an explicit non-local formulation of the classic water wave problem under the long-wave approximation. The normal form analysis is carried out for the newly developed equation, and the associated Benney-Roskes-Davey-Stewartson (BRDS) system which governs the coupled evolution of the envelope of the carrier wave and the wave-induced mean flow is therefore established to predict the existence and stability properties of solitary waves. Numerical results show that three types of free solitary waves exist in the Benney-Luke-type equation as counterparts of the BRDS solutions. They are linked together by a dimension-breaking bifurcation where plane solitary waves and lumps (i.e. fully localised travelling waves in three dimensions) can be viewed as two limiting cases while transversally periodic solitary waves serve as intermediate states. We also study the stability of solitary waves subject to transversal perturbations. The non-elastic behaviour between two stable interacting lumps is investigated by numerical experiments. When the problem is forced by a localised moving load, there exists a a trans-critical regime of forcing speed of which there are no steady solutions, and periodic shedding of lumps can be observed instead.