MODULATION INSTABILITY AND RECURRENCE PHENOMENA IN ANHARMONIC LATTICES

Modulation instability (MI) of running and standing acoustic waves in the Fermi-Pasta-Ulam (FPU) lattice and running optic waves in the Klei n-Gordon (KG) lattice have been investigated analytically and numerica lly. An instability region in wave vector space of perturbation waves additional to that in continuous systems has been found for both FPU a nd KG lattices. The additional instability region takes place for both positive and negative quartic anharmonicity and corresponds to genera tion of backward running perturbation waves. The MI is found to be mor e intense for running carrier waves than that for standing carrier wav es in the FPU lattice, and the MI region for standing waves at the mid dle of the Brillouin zone appears to be strongly asymmetric: It corres ponds to generation of the single side satellite k- /Q/ only. A criter ion for the long-wave modulation instability of running waves which ca n be regarded as a generalization of the Lighthill criterion is propos ed. The influence of the cubic anharmonicity on MI in the FPU lattice has been also investigated. Some specific features of MI-driven recurr ences in discrete systems are pointed out.