Standard Form and a Method of Averaging for Strongly Nonlinear Oscillatory Dispersive Traveling Waves

A method of averaging that is equivalent to but different from the method of multiple scales is introduced to analyze perturbations to strongly nonlinear partial differential equations with oscillatory solutions. Standard form is derived for the amplitude and phase (angle) corresponding to a slowly varying traveling wave satisfying the usual nonlinear dispersion relation. The standard form is linear in spatial and temporal derivatives of the wavenumber and frequency, a nontrivial generalization of standard form for ordinary differential equations. A near identity transformation is determined that eliminates secular terms. To leading order after averaging, conservation of waves and the wave action equation are obtained. Higher-order terms that determine the phase shift are more complicated, but are shown to simplify for purely dissipative perturbations.