Justification of the log-KdV equation in granular chains: the case of precompression

TL;DR

This paper rigorously justifies the use of the log-KdV equation as an approximation for traveling waves in granular chains with precompression, providing error estimates and stability analysis.

Contribution

It establishes the validity of the log-KdV equation as a leading approximation for granular chains with Hertzian potential, including error control and stability results.

Findings

Error bounds for traveling wave approximations

Control of approximation errors over finite time intervals

Nonlinear stability of traveling waves

Abstract

For travelling waves with nonzero boundary conditions, we justify the logarithmic Korteweg-de Vries equation as the leading approximation of the Fermi-Pasta-Ulam lattice with Hertzian nonlinear potential in the limit of small anharmonicity. We prove control of the approximation error for the travelling solutions satisfying differential advance-delay equations, as well as control of the approximation error for time-dependent solutions to the lattice equations on long but finite time intervals. We also show nonlinear stability of the travelling waves on long but finite time intervals.