Initial-value problems for nonlinear diatomic chains

Abstract

An analysis is made of the response of infinite diatomic chains to an initial velocity disturbance. An integral transform solution is obtained for the case of linear interaction force between neighboring particles. Restrictions on the form of the initial condition are indicated for a long-wavelength acoustic response. For nonlinear interactions between neighboring particles, a Korteweg-de Vries equation is obtained for the farfield response of each typical particle within a unit cell, with initial conditions for the Korteweg-de Vries equations obtained by matching to a near-field solution. Numerical solutions are obtained for the standingwave response to a spatially periodic initial condition of harmonic form, with solutions of the chain equations of motion compared to solutions of the Korteweg-de Vries equation. A close correspondence is shown between the two methods of solution for both particle velocity response and the discontinuous strains that occur in a unit cell of a general diatomic chain.