  In this paper we consider the decay rate of solitary-wave solutions to some
classes of non-linear and non-local dispersive equations, including for example
the Whitham equation and a Whitham--Boussinesq system. The dispersive term is
represented by a Fourier multiplier operator that has a real analytic symbol,
and we show that all supercritical solitary-wave solutions decay exponentially,
and moreover provide the exact decay rate, which in general will depend on the
speed of the wave. We also prove that solitary waves have only one crest and
are symmetric for some class of equations.
