  In this paper we study solitary traveling wave solutions to a damped shallow
water system, which is in general quasilinear and of mixed type. We develop a
small data well-posedness theory and prove that traveling wave solutions are a
generic phenomenon that persist with and without viscosity or surface tension
and for all nontrivial traveling wave speeds, even when the parameters dictate
that the equations are hyperbolic and have a sound speed. This theory is
developed by way of a Nash-Moser implicit function theorem, which allows us to
prove strong norm continuity of solutions with respect to the data as well as
the parameters, even in the vanishing limits of viscosity and surface tension.
