Parametrically excited solitary waves

Abstract

A modulated cross-wave of resonant frequencyω₁, carrier frequencyω =ω₁ {1 + O(ε)}, slowly varying complex amplitude O(ε^½b), longitudinal scale b/ε^½ and timescale 1/εω is induced in a long channel of breadth b that contains water of depth d and is subjected to a vertical oscillation of amplitude O(εb) and frequency 2ω, where 0 < ε [Lt ] 1. The complex amplitude satisfies a cubic Schrödinger equation, generalized to incorporate weak damping and the parametric excitation. A solution is obtained that describes the standing solitary wave observed by Wu, Keolian & Rudnick (1984). The results depend on both d/b and l_*/b, where l_* is the capillary length (l_* = 2.7 mm for clean water), and solitary waves are impossible if d/b < 0.325 for l_*/b = 0 or if l_*/b > 0.045 for d/b [gsim ] 1. The corresponding cnoidal waves (of which the solitary wave is a limiting case) are considered in an appendix.