Asymptotic dynamics of short waves in nonlinear dispersive models

Abstract

Multiple-scale perturbation theory, well known for long waves, is extended to the study of the far-field behavior of short waves, commonly called ripples. It is proved that the Benjamin--Bona--Mahony--Peregrine equation can support the propagation of short waves. This result contradicts the Benjamin hypothesis that short waves do not propagate in this model and closes a part of the old controversy over different solutions for the Korteweg--de Vries and Benjamin--Bona--Mahony--Peregrine equations. We have shown that, in a short-wave analysis, a nonlinear (quadratic) Klein-Gordon--type equation replaces the ubiquitous Korteweg--de Vries equation of the long-wave approach. Moreover, the kink solutions of ${\ensuremath{\varphi}}^{4}$ and sine-Gordon equations are understood as an asymptotic behavior of short waves to all orders. It is proved that the antikink solution oF the ${\ensuremath{\varphi}}^{4}$ model, which was never obtained perturbatively, occurs as a perturbation expansion in the wave number $k$ in the short-wave limit.