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

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