  Here we show that asymmetric fully-localized flexural-gravity lumps can
propagate on the surface of an inviscid and irrotational fluid covered by a
variable-thickness elastic material, provided that the thickness varies only in
one direction and has a local minimum. We derive and present equations
governing the evolution of the envelope of flexural-gravity wave packets
allowing the flexing material to have small variations in the transverse (to
propagation) direction. We show that the governing equation belongs to the
general family of Davey-Stewartson equations, but with an extra term in the
surface evolution equation that accounts for the variable thickness of the
elastic cover. We then use an iterative Newton-Raphson scheme, with a numerical
continuation procedure via Lagrange interpolation, in a search to find
fully-localized solutions of this system of equations. We show that if the
elastic sheet thickness has (at least) a local minimum, flexural-gravity lumps
can propagate near the minimum thickness, and in general have an asymmetric
bell-shape in the transverse to the propagation direction. In applied physics,
flexural-gravity waves describe for instance propagation of waves over the
ice-covered bodies of water. Ice is seldom uniform, nor is the seafloor, and in
fact near the boundaries (ice-edges, shorelines) they typically vary only in
one direction (toward to edge), and are uniform in the transverse direction.
This research suggests that fully localized waves are not restricted to
constant ice-thickness/water-depth areas and can exist under much broader
conditions. Presented results may have implications in experimental generation
and observation of flexural-gravity (as well as capillary-gravity) lumps.
