Abstract
This paper addresses the equations of capillary-gravity waves in a two-dimensional channel of finite or infinite depth. These equations are considered in the framework of Hamiltonian systems, for which the Hamiltonian energy has a convergent Taylor expansion in canonical variables near the equilibrium solution. We give an analysis of the Birkhoff normal form transformation that eliminates third-order non-resonant terms of the Hamiltonian. We also provide an analysis of the dynamics of remaining resonant triads in certain cases, related to Wilton ripples.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Alazard, T., Delort, J.-M.: Global solutions and asymptotic behavior for two dimensional gravity water waves. (2013). arXiv:1305.4090. Preprint
Chow, C., Henderson, D., Segur, H.: A generalized stability criterion for resonant triad interactions. J. Fluid Mech. 319, 67–76 (1996)
Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)
Craig, W., Sulem, C., Sulem, P.-L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5, 497–522 (1992)
Craig, W., Worfolk, P.: An integrable normal form for water waves in infinite depth. Physica D 84, 515–531 (1994)
Düll, W.P., Schneider, G., Wayne, C.E.: Justification of the non- linear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth (2013). Preprint
Dyachenko, A.I., Zakharov, V.E.: A dynamic equation for water waves in one horizontal dimension, Eur. J. Mech. B/Fluids 32, 17–21 (2012)
Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. 175, 691–754 (2012)
Hammack, J., Henderson, D.: Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25, 55–97 (1993). Annual Reviews, Palo Alto, CA
Ionescu, A., Pusateri, F.: Global solutions for the gravity water waves system in 2d. (2013). arXiv:1303.5357v2. Preprint
Nazarenko, S.: Wave Turbulence. Lecture Notes in Physics, vol. 825. Springer, Heidelberg (2011). xvi+279 pp. ISBN: 978-3-642-15941-1
Totz, D., Wu, S.: A rigorous justification of the modulation approximation to the 2D full water wave problem. Commun. Math. Phys. 310, 817–883 (2012)
Wu, S.: Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177, 45–135 (2009)
Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 1990–1994 (1968)
Zakharov, V.E.: Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid, Eur. J. Mech. B/Fluids 18, 327–344 (1999)
Zakharov, V.E., L’vov, V.S., Falkovich, G.: Komogorov Spectra of Turbulence. Springer, Berlin (1992)
Acknowledgements
WC is partially supported by the Canada Research Chairs Program and NSERC through grant number 238452–11. CS is partially supported by NSERC through grant number 46179–13 and Simons Foundation Fellowship 265059. CS would like to extend her warmest wishes to Walter on the occasion of his 60th birthday.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Craig, W., Sulem, C. (2015). Normal Form Transformations for Capillary-Gravity Water Waves. In: Guyenne, P., Nicholls, D., Sulem, C. (eds) Hamiltonian Partial Differential Equations and Applications. Fields Institute Communications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2950-4_3
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2950-4_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2949-8
Online ISBN: 978-1-4939-2950-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)