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Six wave interaction equations in finite-depth gravity waves with surface tension

Published online by Cambridge University Press:  14 April 2023

Mark J. Ablowitz ,
Xu-Dan Luo Open the ORCID record for Xu-Dan Luo [Opens in a new window]  and
Ziad H. Musslimani Open the ORCID record for Ziad H. Musslimani [Opens in a new window]
Mark J. Ablowitz
Affiliation:
Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, CO 80309-0526, USA
Xu-Dan Luo
Affiliation:
Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China
Ziad H. Musslimani*
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA
*
†Email address for correspondence: musslimani@math.fsu.edu
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Abstract

Three wave resonant triad interactions in two space and one time dimensions form a well-known system of first-order quadratically nonlinear evolution equations that arise in many areas of physics. In deep water waves, they were first derived by Simmons in 1969 and later shown to be exactly solvable by Ablowitz & Haberman in 1975. Specifically, integrability was established by introducing a system of six wave interactions whose symmetry reduction leads to the well-known three wave equations. Here, it is shown that the six wave interaction and classical three wave equations satisfying triad resonance conditions in finite-depth gravity waves can be derived from the non-local integro-differential formulation of the free surface gravity wave equation with surface tension. These quadratically nonlinear six wave interaction equations and their reductions to the classical and non-local complex as well as real reverse space–time three wave interaction equations are integrable. Limits to infinite and shallow water depth are also discussed.

JFM classification

Mathematical Foundations: Hamiltonian theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 961 , 25 April 2023 , A3
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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