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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: [email protected]
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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

REFERENCES

Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48 (4), 377–385.CrossRefGoogle Scholar
Brinch-Nielsen, U. & Jonsson, I. G. 1986 Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth. Wave Motion 8 (5), 455–472.CrossRefGoogle Scholar
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29 (6), 430–446.CrossRefGoogle Scholar
Caponi, E. A., Saffman, P. G. & Yuen, H. C. 1982 Instability and confined chaos in a non-linear dispersive wave system. Phys. Fluids 25 (12), 2159–2166.CrossRefGoogle Scholar
Carter, J. D. 2001 Stability and existence of traveling wave solutions of the two-dimensional nonlinear Schrödinger equation and its higher-order generalizations. PhD thesis, University of Colorado at Boulder, CO.Google Scholar
Crawford, D. R., Saffman, P. G. & Yuen, H. C. 1980 Evolution of a random inhomogeneous field of non-linear deep-water gravity waves. Wave Motion 2, 1–16.CrossRefGoogle Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338 (1613), 101–110.Google Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369 (1736), 105–114.Google Scholar
Fenton, J. D. 1985 A fifth-order Stokes theory for steady waves. J. Waterway Port Coastal Ocean Engng 111 (2), 216–234.CrossRefGoogle Scholar
Gorman, R. M. 2003 The treatment of discontinuities in computing the nonlinear energy transfer for finite-depth gravity wave spectra. J. Atmos. Ocean. Technol. 20 (1), 206–216.2.0.CO;2>CrossRefGoogle Scholar
Gramstad, O. & Trulsen, K. 2010 Can swell increase the number of freak waves in a wind sea? J. Fluid Mech. 650, 57–79.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2009 On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J. Fluid Mech. 637, 1–44.CrossRefGoogle Scholar
Janssen, P. A. E. M. & Onorato, M. 2007 The intermediate water depth limit of the Zakharov equation and consequences for wave prediction. J. Phys. Oceanogr. 37, 2389–2400.CrossRefGoogle Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Krasitskii, V. P. & Kalmykov, V. A. 1993 Four-wave reduced equations for surface gravity waves. Izv. Atmos. Ocean. Phys. 29 (2), 222–228.Google Scholar
Kuznetsov, E. A. 1977 Solitons in a parametrically unstable plasma. Dokl. USSR 236, 575–577.Google Scholar
Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–416.CrossRefGoogle Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.CrossRefGoogle Scholar
McLean, J. W. 1982 Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.CrossRefGoogle Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.CrossRefGoogle Scholar
Peregrine, D. H. 1983 Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16–43.CrossRefGoogle Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.CrossRefGoogle Scholar
Segur, H., Henderson, D. M. & L., Hammack, J. 2005 Can the Benjamin–Feir instability spawn a rogue wave? In Proceedings of the 14th 'Aha Huliko'a Hawaiian Winter Workshop, Honolulu, Hawaii (ed. Müller, P. & Henderson, D.), pp. 43–57. SOEST Special Publication.Google Scholar
Stiassnie, M. 1984 Note on the modified nonlinear Schrödinger-equation for deep-water waves. Wave Motion 6 (4), 431–433.CrossRefGoogle Scholar
Stiassnie, M. & Gramstad, O. 2009 On Zakharov's kernel and the interaction of non-collinear wavetrains in finite water depth. J. Fluid Mech. 639, 433–442.CrossRefGoogle Scholar
Stiassnie, M. & Shemer, L. 1984 On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143, 47–67.CrossRefGoogle Scholar
Sulem, C. & Sulem, P. L. 1999 The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, vol. 139. Springer.Google Scholar
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24 (3), 281–289.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1997 Frequency downshift in three-dimensional wave trains in a deep basin. J. Fluid Mech. 352, 359–373.CrossRefGoogle Scholar
Trulsen, K., Kliakhandler, I., Dysthe, K. B. & Velarde, M. G. 2000 On weakly nonlinear modulation of waves on deep water. Phys. Fluids 12 (10), 2432–2437.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
Zakharov, V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. B 18 (3), 327–344.CrossRefGoogle Scholar
Zakharov, V. E. & Dyachenko, A. I. 2010 About shape of giant breather. Eur. J. Mech. B 29 (2), 127–131.CrossRefGoogle Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69.Google Scholar