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On a new non-local formulation of water waves

Published online by Cambridge University Press:  14 August 2006

M. J. ABLOWITZ ,
A. S. FOKAS  and
Z. H. MUSSLIMANI
M. J. ABLOWITZ
Affiliation:
Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO 80309-0526, USA
A. S. FOKAS
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Z. H. MUSSLIMANI
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA
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Abstract

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The classical equations of water waves are reformulated as a system of two equations, one of which is an explicit non-local equation, for the wave height and for the velocity potential evaluated on the free surface. Evaluation of the velocity potential as a function of the depth is not required in order to calculate the wave height and the velocity potential on the free surface. The non-local system yields integral relations related to mass and centre of mass, and is shown to reduce to known asymptotic limits in shallow and deep water. Included in these asymptotic reductions are the Boussinesq, Benney–Luke and nonlinear Schrödinger equations. Two-dimensional lumps with sufficient surface tension are obtained numerically. The extension of this non-local formulation to the case of a variable bottom is also presented.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 562 , 10 September 2006 , pp. 313 - 343
Copyright
© 2006 Cambridge University Press