In this talk, we study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa-Holm equation and its generalizations. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a variety of numerical methods to study both the analytical and physical properties of the Camassa-Holm equation and show its potential for modeling the propagation of tsunami waves. In particular, we provide global existence and uniqueness results for the Camassa-Holm Equation by establishing convergence results for the particle method applied to these equations, and then use this same method to numerically quantify the nonlinear interaction among the peakon solutions. We conclude the talk by proposing new invariant-preserving finite difference schemes for a generalization of the Camassa-Holm equation as a potential model for the propagation of tsunami waves.