Two-dimensional wave packets in an elastic solid with couple stresses

Abstract

The problem of (2+1) (two spatial and one temporal) dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. The aim is to derive (2+1) non-linear model equations for the description of elastic waves in the far field. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that the modulation of waves is governed by a system of three non-linear evolution equations. These equations involve amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave, and will be called the “generalized Davey–Stewartson equations”. Under some restrictions on parameter values, the generalized Davey–Stewartson equations reduce to the Davey–Stewartson and to the non-linear Schrödinger equations. Finally, some special solutions involving sech–tanh–tanh and tanh–tanh–tanh type solitary wave solutions are presented.