Parametrically excited solitary waves

A modulated cross-wave of resonant frequencyω 1 , carrier frequencyω =ω 1 {1 + O (e)}, slowly varying complex amplitude O (e ½ b ), longitudinal scale b /e ½ and timescale 1/eω 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 (e b ) and frequency 2ω, where 0 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 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.