Dynamics of nonlinear mass-spring chains near the continuum limit

Abstract

We study the dynamics of mass-spring chains with arbitrary interparticle and substrate potentials and describe a systematic approach to derive the equations of motion near the continuum limit. Our method, which applies both to strictly one-dimensional problems and to one-dimensional chains free to move in three dimensions, correctly captures all terms to given order in discreteness and allows us to formulate well-behaved nonlinear partial differential equations for these systems. We re-examine the familiar cases of the Fermi-Pasta-Ulam problem and the Frenkel-Kontorova model and obtain new insights into these problems.