From Newton's cradle to the discrete p-Schr\"odinger equation

TL;DR

This paper analyzes the dynamics of nonlinear oscillator chains, deriving a discrete p-Schr"odinger equation as an envelope approximation, and demonstrates the existence of long-lived breather solutions and decay estimates for small amplitude states.

Contribution

It provides a rigorous asymptotic derivation of the discrete p-Schr"odinger equation for nonlinear oscillator chains, including Newton's cradle, and proves the existence of breather solutions.

Findings

Existence of long-lived breather solutions.

Derivation of the discrete p-Schr"odinger equation as an envelope model.

Decay estimates for small amplitude solutions over long times.

Abstract

We investigate the dynamics of a chain of oscillators coupled by fully-nonlinear interaction potentials. This class of models includes Newton's cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equation leading to approximate solutions is a discrete p-Schr\"odinger equation. Our results include the existence of long-lived breather solutions to the original model. For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times.