  The dynamics of solitary gravity-capillary water waves propagating on the
surface of a three-dimensional fluid domain is studied numerically. In order to
accurately compute complex time dependent solutions, we simplify the full
potential flow problem by taking a cubic truncation of the scaled
Dirichlet-to-Neumann operator for the normal velocity on the free surface. This
approximation agrees remarkably well with the full equations for the
bifurcation curves, wave profiles and the dynamics of solitary waves for a
two-dimensional fluid domain. Fully localised solitary waves are then computed
in the three-dimensional problem and the stability and interaction of both line
and localized solitary waves are investigated via numerical time integration of
the equations. The solitary wave branches are indexed by their finite energy at
small amplitude, and the dynamics of the solitary waves is complex involving
nonlinear focussing of wave packets, quasi-elastic collisions, and the
generation of propagating, spatially localised, time-periodic structures
(breathers).
