Variation of Wave Action: Modulations of the Phase Shift for Strongly Nonlinear Dispersive Waves with Weak Dissipation. A New Adiabatic Invariant Involving the Modulated Phase Shift for Strongly Nonlinear, Slowly Varying, and Weak

摘要

The equations for the spatial and temporal modulations of the phase shift for slowly varying strongly nonlinear oscillators and dispersive waves have been determined for the first time. The effects of dissipative perturbations have been investigated for nonlinear oscillatory solutions of ordinary and partial differential equations (described by Klein-Gordon and Korteweg-de Vries type equations). The phase shift equations were derived using the method of multiple scales by evaluating the small perturbations to the exact action equation, a somewhat simpler technique than usual elimination of secular terms at an even higher order in the asymptotic expansion. It has been shown that, for dissipative perturbations, the frequency and action equations are valid to higher order and that their variations are only due to perturbations in the wave number and the averaged amplitude parameters. For second-order ordinary differential equations, the phase shift is determined from initial conditions in straight-forward manner since it was shown that there exists a new adiabatic invariant.