  Hydroelastic solitary waves propagating on the surface of a three-dimensional
ideal fluid through the deformation of an elastic sheet are studied. The
problem is investigated based on a Benney-Luke-type equation derived via an
explicit non-local formulation of the classic water wave problem. The normal
form analysis is carried out for the newly developed equation, which results in
the Benney-Roskes-Davey-Stewartson (BRDS) system governing the coupled
evolution of the envelope of a carrier wave and the wave-induced mean flow.
Numerical results show three types of free solitary waves in the
Benney-Luke-type equation: plane solitary wave, lump (i.e., fully localized
traveling waves in three dimensions), and transversally periodic solitary wave,
and they are counterparts of the BRDS solutions. They are linked together by a
dimension-breaking bifurcation where plane solitary waves and lumps can be
viewed as two limiting cases, and transversally periodic solitary waves serve
as intermediate states. The stability and interaction of solitary waves are
investigated via a numerical time integration of the Benney-Luke-type equation.
For a localized load moving on the elastic sheet with a constant speed, it is
found that there exists a transcritical regime of forcing speed for which there
are no steady solutions. Instead, periodic shedding of lumps can be observed if
the forcing moves at speed in this range.
