Existence theory of capillary-gravity waves on water of finite depth

Shu-Ming Sun

doi:10.3934/mcrf.2014.4.315

Article Contents

2014, Volume 4, Issue 3: 315-363. Doi: 10.3934/mcrf.2014.4.315

This issue Previous Article Time optimal control problems for some non-smooth systems Next Article Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities

Existence theory of capillary-gravity waves on water of finite depth

Shu-Ming Sun^1,

1.
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

Received: February 28, 2013

Revised: December 31, 2013

Published: August 2014

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Abstract

This review article discusses the recent developments on the existence of two-dimensional and three-dimensional capillary-gravity waves on water of finite-depth. The Korteweg-de Vries (KdV) equation and Kadomtsev-Petviashvili (KP) equation are derived formally from the exact governing equations and the solitary-wave solutions and other solution are obtained for these model equations. Recent results on the existence of solutions of the exact governing equations near the solutions of these model equations are presented and various two- and three-dimensional solutions of the exact equations are provided. The ideas and methods to obtain the existence results are briefly discussed.

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Mathematics Subject Classification: Primary: 76B15, 35Q31; Secondary: 76B25.

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