Localized Modes in the Long-Time Behavior of Anharmonic Lattices

The time evolution of pure anharmonic lattices with quartic, positive lattice anharmonicity is studied by transforming nonlinear differential-difference equations into nonlinear integro-difference equations with kernels given by lattice Green's functions. Such a formulation enables us to treat one-, two-, and three-dimensional lattices on equal footing. By studying the asymptotic properties for t →∞ of the equations, it is shown that a long-lived, spatially localized oscillatory mode can exist under certain conditions for each of these three cases. A quasi-nonergodic behavior of anharmonic lattices obtained here may be of different nature from that found by Fermi, Pasta, and Ulam.