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