Nonresonant interacting waves for the nonlinear Klein–Gordon equation in three-dimensional space

Interaction among nonresonant waves of the nonlinear Klein–Gordon equation in ordinary (three-dimensional) space is investigated, by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. We show that the slow amplitude modulation of Fourier modes can be described by a system of nonlinear evolution equations. The system is C-integrable, i.e. can be linearized through an appropriate transformation of the dependent variables. N-period quasiperiodic solutions with a nonlinear dispersion relation are observed. Moreover, envelope solitons with fixed but arbitrary shapes and velocities connected to the group velocities of the carrier waves are possible. During a collision, solitons maintain their shape, but are subjected to a phase shift. The technique proposed in this paper can be applied to the description of soliton interactions in nonlinear dispersive media without using the complexity of the inverse scattering method.