Normal Form Transformations for Capillary-Gravity Water Waves
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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.
KeywordsNormal Form Transformation Capillary-gravity Water Waves Resonant Triad Wilton Ripples Polar Coordinate Singularities
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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