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VARIATIONAL PRINCIPLES FOR WATER WAVES FROM THE VIEWPOINT OF A TIME‐DEPENDENT MOVING MESH
Author(s) -
Bridges Thomas J.,
Donaldson Neil M.
Publication year - 2011
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579310001233
Subject(s) - mathematics , conservation law , partial differential equation , hamiltonian (control theory) , nonlinear system , laplace operator , mathematical analysis , mathematical optimization , physics , quantum mechanics
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.