In this paper we first present a short survey of the mathematical models of tsunami waves. More precisely, we investigate the traveling wave solutions of tsunami model which is described by a system of nonlinear partial differential equations (PDE) with shallow water conditions. We obtain peakons, cuspon-peakon solutions and we present their geometrical interpretation through computer simulations. We approximate the tsunami model via cellular nonlinear networks and study the profiles of the traveling wave solutions. Computer simulations illustrate the obtained theoretical results.