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 ${\phi }^4$ 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 ${\phi }^4$ model, which was never obtained perturbatively, occurs as a perturbation expansion in the wave number $k$ in the short-wave limit.

• Received 19 August 1997

DOI:https://doi.org/10.1103/PhysRevE.57.6206