A Plethora of Solitary Gravity-Capillary Water Waves with Nearly Critical Bond and Froude Numbers

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 354; no. 1707; pp. 575 - 607
• Main Authors: Buffoni, B., Groves, M. D., Toland, John Francis
• Format: Journal Article
• Language: English
• Published: London The Royal Society 15.03.1996
• Subjects: Boundary conditions; Coordinate systems; Eigenvalues; Froude number; Interfacial tension; Mathematical manifolds; Mathematics; Solitons; Water waves; Waves
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.1996.0020

This paper considers the existence of solitary-wave solutions of the classical waterwave problem in the presence of surface tension. A region of Bond number-Froude number parameter space close to (1/3, 1) is identified, at each point of which there are infinitely many distinct multi-troughed solitary waves of depression. The method is to study a Hamiltonian formulation of the mathematical problem for solitary waves using a centre-manifold technique valid near Bond number 1/3 and Froude number 1. The problem is thus replaced by an equivalent problem posed on a four-dimensional manifold. In a certain region of parameter space near (1/3, 1), there is a Smale horseshoe in the dynamics on the centre manifold and therefore infinitely many distinct homoclinic orbits.