Nonlinear interaction between long and short waves in a generalized elastic solid

In the present study, nonlinear interaction between long and short waves propagating in a generalized elastic medium is examined. In particular, the case where the phase velocity of the long longitudinal wave is equal to the group velocity of the short transverse waves is studied. By using an asymptotic expansion method, three coupled nonlinear evolution equations are derived for the description of the interaction. In the absence of one of the transverse waves, these equations reduce to the so-called long wave–short wave interaction equations which are also known as Zakharov–Benney (ZB) equations. Furthermore, the nonlinear interaction between a long longitudinal wave and a short longitudinal wave is considered and ZB equations are derived for the description of interaction. Some special solutions to the interaction equations are also presented.