The modulation of weakly nonlinear dispersive waves of an extended Φ4 model near the marginal state of instability

Abstract

Wave modulations in one-dimensional anharmonic lattices are studied by the use of a perturbation method. Each atom is assumed to interact with its first neighbors following a double well interatomic potential, and with its second neighbors following a harmonic potential. The amplitude modulation of an extended Φ4 chain can be described by the nonlinear Schrödinger (NLS) equation and the carrier wave is modulationally unstable for wavenumbers k larger than a critical wavenumber kc.However, when k is near kc the NLS equation is invalid and the modulations are governed by a modified form of the NLS equation that involves higher-order non-linearities. The correct critical wavenumber for the marginal modulational instability is obtained and exact solutions are presented.