The slowly varying phase shift for perturbed, single and multi-phased, strongly nonlinear, dispersive waves (Q1117779)

Slowly varying, strongly nonlinear, dispersive, oscillatory waves are analyzed for equations which may be represented by a Lagrangian, including the effects of perturbations. The linear partial differential equation for the modulations of the phase shift follows from a perturbation analysis of the exact equation for wave action. For purely dissipative perturbations, it is shown that variations of the wave action, its flux, and its dissipation are due to perturbations of the wave number and frequency (if, in addition, the dissipative effect of a higher-order perturbation is included), a result the authors have previously developed for some specific problems. Similar conclusions are also derived for strongly nonlinear dispersive waves with higher spatial dimensional and with multiple oscillatory phases. Furthermore, only the nonhomogeneous terms for the phase shift are altered for more general perturbations.