  The KP-I equation \[ (u_t-2uu_x+\tfrac{1}{2}(\beta-\tfrac{1}{3})u_{xxx})_x
-u_{yy}=0 \] arises as a weakly nonlinear model equation for gravity-capillary
waves with strong surface tension (Bond number $\beta>1/3$). This equation
admits --- as an explicit solution --- a `fully localised' or `lump' solitary
wave which decays to zero in all spatial directions. Recently there has been
interest in the \emph{full-dispersion KP-I equation} \[u_t + m({\mathrm D}) u_x
+ 2 u u_x = 0,\] where $m({\mathrm D})$ is the Fourier multiplier with symbol
\[ m(k) = \left( 1 + \beta |k|^2|\right)^{\frac{1}{2}} \left( \frac{\tanh
|k|}{|k|} \right)^{\frac{1}{2}} \left( 1 + \frac{2k_2^2}{k_1^2}
\right)^{\frac{1}{2}}, \] which is obtained by retaining the exact dispersion
relation from the water-wave problem. In this paper we show that the FDKP-I
equation also has a fully localised solitary-wave solution. The existence
theory is variational and perturbative in nature. A variational principle for
fully localised solitary waves is reduced to a locally equivalent variational
principle featuring a perturbation of the variational functional associated
with fully localised solitary-wave solutions of the {KP-I} equation. A
nontrivial critical point of the reduced functional is found by minimising it
over its natural constraint set.
