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VARIATIONAL PRINCIPLES FOR WATER WAVES FROM THE VIEWPOINT OF A TIME-DEPENDENT MOVING MESH

Part of: Incompressible inviscid fluids Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  17 November 2010

Thomas J. Bridges  and
Neil M. Donaldson
Thomas J. Bridges
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, U.K. (email: t.bridges@surrey.ac.uk)
Neil M. Donaldson
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, U.K.
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Abstract

The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

MSC classification

Secondary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 147 - 173
Copyright
Copyright © University College London 2011

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