Modulation instability and recurrence phenomena in anharmonic lattices

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Abstract
Modulation instability (MI) of running and standing acoustic waves in the Fermi-Pasta-Ulam (FPU) lattice and running optic waves in the Klein-Gordon (KG) lattice have been investigated analytically and numerically. An instability region in wave vector space of perturbation waves additional to that in continuous systems has been found for both FPU and KG lattices. The additional instability region takes place for both positive and negative quartic anharmonicity and corresponds to generation of backward running perturbation waves. The MI is found to be more intense for running carrier waves than that for standing carrier waves in the FPU lattice, and the MI region for standing waves at the middle of the Brillouin zone appears to be strongly asymmetric: It corresponds to generation of the single side satellite k-|Q| only. A criterion for the long-wave modulation instability of running waves which can be regarded as a generalization of the Lighthill criterion is proposed. The influence of the cubic anharmonicity on MI in the FPU lattice has been also investigated. Some specific features of MI-driven recurrences in discrete systems are pointed out. © 1996 The American Physical Society.