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, the linear terms of which match the dispersion relation for gravity water waves on finite depth. 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.