Secondary solitary wave formation in systems with generalized Hertz interactions

We consider a chain of monodisperse elastic grains of radius R where the grains are barely in contact. The grains repel upon contact via the Hertz-type potential, V proportional to delta(n), n > 2, where delta > or = 0, is the grain-grain overlap, delta identical with 2R-(u(i+1)-u(i)), where u(i) denotes the displacement of grain i from its original equilibrium position. This being a computational study, we consider n to be arbitrary. Our dynamical simulations build on several earlier studies by Nesterenko, Coste, and Sen and co-workers that have shown that an impulse propagates as a solitary wave of fixed spatial extent, infinity < L(n) < 1, through a chain of grains. Here, we develop on a recent study by Manciu, Sen, and Hurd [Phys. Rev. E 63, 016614 (2001)] that shows that colliding solitary waves in the chains of interest spawn a well-defined hierarchy of multiple secondary solitary waves (SSWs) that carry approximately 0.5% or less of the energy of the original solitary waves. We show that the emergence of SSWs is a complex process where nonlinear forces and the discreteness of the grains lead to the partitioning of the available energy into hierarchies of SSWs. The process of formation of SSWs involves length scales and time scales that are controlled by the strength of the nonlinearity in the system. To the best of our knowledge, there is no formal theory that describes the dynamics associated with the formation of SSWs. Calculations for cases where the Hertz-type potential can be symmetric in the overlap parameter delta, i.e., where delta can be both positive and negative, suggest that the formation of secondary solitary waves may be a fundamental property of certain discrete, nonlinear systems.