The traveling wave problem for the shallow water equations: well-posedness and the limits of vanishing viscosity and surface tension

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.