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Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth

Published online by Cambridge University Press:  26 January 2011

ODIN GRAMSTAD  and
KARSTEN TRULSEN
ODIN GRAMSTAD
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
KARSTEN TRULSEN*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
*
†Email address for correspondence: karstent@math.uio.no
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Abstract

The commonly used forms of the modified nonlinear Schrödinger equations for deep water (Dysthe, Proc. R. Soc. Lond. A, vol. 369, 1979, p. 105) and arbitrary depth (Brinch–Nielsen & Jonsson, Wave Motion, vol. 8, 1986, p. 455) do not conserve momentum and are not Hamiltonian. We show how these equations can be brought into Hamiltonian form, with the action, momentum and Hamiltonian being conserved. We derive the new fourth-order nonlinear Schrödinger equation for arbitrary depth, starting from the Zakharov equation enhanced with the new kernel of Krasitskii (J. Fluid Mech., vol. 272, 1994, p. 1).

JFM classification

Mathematical Foundations: Hamiltonian theory Waves/Free-surface Flows: Surface gravity waves Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 404 - 426
Copyright
Copyright © Cambridge University Press 2011

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References

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