Stability and motion of intrinsic localized modes in nonlinear periodic lattices

Abstract

Previous theoretical studies and molecular-dynamics simulations show that a periodic one-dimensional lattice with nearest-neighbor quadratic and quartic interactions supports stationary localized modes. The most localized of these are an odd-parity mode with a displacement pattern A(. . . ,0,-1/2,1,-1/2,0, . . .) and an even-parity mode A(. . . ,0,-1,1,0. . .), where A is the amplitude. These solutions are asymptotically exact for the pure even-order anharmonic lattice in the limit of increasing order. We show here that in both this asymptotic limit and for the harmonic plus quartic lattice, the odd-parity mode is unstable to infinitesimal perturbations, while the even-parity mode is stable. For the pure quartic case, the predicted growth rate for the instability is 0.15 in units of the mode frequency, in excellent agreement with the rate observed in our molecular-dynamics simulations. In contrast, we observe the even-parity mode to persist unchanged over more than 32 000 mode oscillations. Our simulations show that the instability does not destroy the odd mode, but causes it to move. We will also discuss a smoothly traveling version of these modes. As they move, these modes have a nonconstant phase difference between adjacent relative displacements, in contrast with traveling modes discussed previously by others.