Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation

TLDR: In this article, a spatially discrete modulation equation for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations was derived, where the fully nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that is called the discrete p-Schroedinger (DpS) equation.

Abstract: We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully-nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schroedinger (DpS) equation. It consists of a spatial discretization of a generalized Schroedinger equation with p-Laplacian, with fractional p>2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic travelling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic travelling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and travelling breathers, and repulsive or attractive interactions of these localized structures.