Justification of the log--KdV Equation in Granular Chains: The Case of Precompression

For traveling 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 traveling wave 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 metastability of the traveling waves on long but finite time intervals.