Parametrically excited solitary waves

A modulated cross-wave of resonant frequencyω 1 , carrier frequencyω =ω 1 {1 + O (ε)}, slowly varying complex amplitude O (ε ½ b ), 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 (ε b ) 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.