Boundary effects on the dynamics of chains of coupled oscillators

TL;DR

This paper investigates how boundary conditions affect the long-wavelength, small-amplitude dynamics of coupled oscillator chains, showing that solutions approximate nonlinear Schrödinger equations with boundary-dependent decay properties.

Contribution

It provides a rigorous analysis of boundary effects on the approximation of Klein-Gordon lattice dynamics by nonlinear Schrödinger equations, highlighting differences between periodic and Dirichlet boundaries.

Findings

Main solution approximated by nonlinear Schrödinger equation with exponential decay of Fourier coefficients.

First order correction decays exponentially for periodic boundaries, but only as a power law for Dirichlet boundaries.

Results explain numerical observations from previous studies.

Abstract

We study the dynamics of a chain of coupled particles subjected to a restoring force (Klein-Gordon lattice) in the cases of either periodic or Dirichlet boundary conditions. Precisely, we prove that, when the initial data are of small amplitude and have long wavelength, the main part of the solution is interpolated by a solution of the nonlinear Schr\"odinger equation, which in turn has the property that its Fourier coefficients decay exponentially. The first order correction to the solution has Fourier coefficients that decay exponentially in the periodic case, but only as a power in the Dirichlet case. In particular our result allows one to explain the numerical computations of the paper \cite{BMP07}.