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

Eric Dumas (1) , Dmitry E. Pelinovsky (2)

1 IF - Institut Fourier
2 Department of Mathematics and Statistics [Hamilton]

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.

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Formation de Structures et Solitons [nlin.PS]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Eric Dumas, Dmitry E. Pelinovsky. Justification of the log-KdV equation in granular chains: the case of precompression. SIAM Journal on Mathematical Analysis, 2014, 46 (6). ⟨hal-01003727⟩