Vibration of a Chain with Nonlinear Interaction

Abstract

Vibration of a chain of particles interacting by nonlinear force is investigated. Using a transformation exact solutions to the equation of motion are aimed at. For a special type of interaction potential of the form \begin{aligned} \phi(r){=}\frac{a}{b}e^{-br}+ar+\text{const.},\ (a,b{>}0) \end{aligned} exact solutions are actually obtained in terms of the Jacobian elliptic functions. It is shown that the system has N “normal modes”. Expansion due to vibration or “thermal expansion” of the chain is also discussed.