Nonlinear modulation of multidimensional lattice waves

The equations governing weakly nonlinear modulations of N-dimensional lattices are considered using a quasidiscrete multiple-scale approach. It is found that the evolution of a short wave packet for a lattice system with cubic and quartic interatomic potentials is governed by the generalized Davey-Stewartson (GDS) equations, which include mean motion induced by the oscillatory wave packet through cubic interatomic interaction. The GDS equations derived here are more general than those known in the theory of water waves because of the anisotropy inherent in lattices. The generalized Kadomtsev-Petviashvili equations describing the evolution of long-wavelength acoustic modes in two- and three-dimensional lattices are also presented. Then the modulational instability of an N-dimensional Stokes lattice wave is discussed based on the N-dimensional GDS equations obtained. Finally, the one- and two-soliton solutions of two-dimensional GDS equations are provided by means of Hirota’s bilinear transformation method.