  In this paper we address the return to equilibrium problem for an
axisymmetric floating structure in shallow water. First we show that the
equation for the solid motion can be reduced to a delay differential equation
involving an extension-trace operator whose role is to describe the influence
of the fluid equations on the solid motion. It turns out that the compatibility
conditions on the initial data for the return to equilibrium configuration are
not satisfied, so we cannot use the result from [3] for the nonlinear problem.
Hence, assuming small amplitude waves, we linearize the equations in the
exterior domain and we keep the nonlinear equations in the interior domain. For
such configurations, the extension-trace operator can be computed explicitly
and the delay term in the differential equation can be put in convolution form.
The solid motion is therefore governed by a nonlinear second order
integro-differential equation, whose linearization is the well-known Cummins
equation. We show global in time existence and uniqueness of the solution using
the conservation of the total fluid-structure energy.
