  We prove existence of small-amplitude modulated solitary waves for the
full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface
tension. The resulting waves are small-order perturbations of scaled,
translated and frequency-shifted solutions of a Davey--Stewartson (DS) type
equation. The construction is variational and relies upon a series of reductive
steps which transform the FDKP functional to a perturbed scaling of the DS
functional, for which least-energy ground states are found. We also establish a
convergence result showing that scalings of FDKP solitary waves converge to
ground states of the DS functional as the scaling parameter tends to zero.
  Our method is robust and applies to nonlinear dispersive equations with the
properties that (i) their dispersion relation has a global minimum (or maximum)
at a non-zero wave number, and (ii) the associated formal weakly nonlinear
analysis leads to a DS equation of elliptic-elliptic focussing type. We present
full details for the FDKP equation.
