E. Dumas and D. Pelinovsky

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

SIAM Journal of Mathematical Analysis 46, 4075-4103 (2014)

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.

Keywords:
Discrete FPU lattice, granular crystals, log-KdV equation, existence and stability, justification of amplitude equations.