  We study the existence of traveling wave solutions to a unidirectional
shallow water model which incorporates the full linear dispersion relation for
both gravitational and capillary restoring forces. Using functional analytic
techniques, we show that for small surface tension (corresponding to Bond
numbers between $0$ and ${1}/{3}$) there exists small amplitude solitary waves
that decay to asymptotically small periodic waves at spatial infinity. The size
of the oscillations in the far field are shown to be small beyond all algebraic
orders in the amplitude of the wave. We also present numerical evidence, based
on the recent analytical work of Hur \& Johnson, that the asymptotic end states
are modulationally stable for all Bond numbers between $0$ and $1/3$.
