We consider 1D systems of masses, which can transfer energy via harmonic and/or anharmonic interactions of the form V(xi,i+1) ∼ x i,i+1β, where β > 2, and where the potential energy is physically meaningful. The systems are placed within boundaries or satisfy periodic boundary conditions. Any velocity perturbation in these (non-integrable) systems is found to travel as discrete solitary waves. These solitary waves very nearly preserve themselves and make tiny secondary solitary waves when they collide or reach a boundary. As time t → ∞, these systems cascade to an equilibrium-like state, with Boltzmann-like velocity distributions, yet with no equipartitioning of energy, which we refer to and briefly describe as the "quasi-equilibrium" state. © 2004 Elsevier B.V. All rights reserved.