  We consider a class of pseudodifferential evolution equations of the form
$$u_t + (n(u) + Lu)_x = 0,$$ in which $L$ is a linear smoothing operator and
$n$ is at least quadratic near the origin; this class includes in particular
the Whitham equation. A family of solitary-wave solutions is found using a
constrained minimisation principle and concentration-compactness methods for
noncoercive functionals. The solitary waves are approximated by (scalings of)
the corresponding solutions to partial differential equations arising as weakly
nonlinear approximations; in the case of the Whitham equation the approximation
is the Korteweg-deVries equation. We also demonstrate that the family of
solitary-wave solutions is conditionally energetically stable.
