Mathematical Control and Related Fields (MCRF)

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

Pages: 315 - 363, Volume 4, Issue 3, September 2014      doi:10.3934/mcrf.2014.4.315

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Shu-Ming Sun - Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, United States (email)

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.

Keywords:  Capillary-gravity waves, water waves, existence theory.
Mathematics Subject Classification:  Primary: 76B15, 35Q31; Secondary: 76B25.

Received: March 2013;      Revised: January 2014;      Available Online: April 2014.