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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 particularthe Whitham equation. A family of solitary-wave solutions is found using aconstrained minimisation principle and concentration-compactness methods fornoncoercive functionals. The solitary waves are approximated by (scalings of)the corresponding solutions to partial differential equations arising as weaklynonlinear approximations; in the case of the Whitham equation the approximationis the Korteweg-deVries equation. We also demonstrate that the family ofsolitary-wave solutions is conditionally energetically stable.

On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type