Normal Form Transformations for Capillary-Gravity Water Waves

  • Walter Craig
  • Catherine SulemEmail author
Part of the Fields Institute Communications book series (FIC, volume 75)


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.


Normal Form Transformation Capillary-gravity Water Waves Resonant Triad Wilton Ripples Polar Coordinate Singularities 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.The Fields InstituteTorontoCanada
  2. 2.Department of MathematicsMcMaster UniversityHamiltonCanada
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

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