The sub-harmonic bifurcation of Stokes waves
type: Item
instance of: scholarly article
title: The sub-harmonic bifurcation of Stokes waves
zbMATH Open document ID: 0962.76012
published in: Archive for Rational Mechanics and Analysis
publication date: 2000-07-09T00:00:00Z
review text: The behaviour of steady periodic water waves on water of infinite depth, that satisfy exactly the kinematic and dynamic boundary conditions on the free surface of water, with or without surface tension, are given by solutions of a nonlinear pseudo-differential operator equation for a \(2\pi\)-periodic function of a real variable. The study is complicated by the fact that the equation is quasilinear, and it involves a non-local operator in the form of a Hilbert transform. Although this equation is exact, it is quadratic with no higher order terms, and the global structure of its solution set can be studied using elements of the theory of real analytic varieties, and using the variational technique. The purpose of this paper is to show that uniquely defined arc-wise connected set of solutions with prescribed minimal period bifurcates from the first eigenvalue of the linearized problem. Although the set is not necessarily maximal as a connected set of solutions, and may possibly self-intersect, it has a local real analytic parametrization that contains a wave of greatest height in its closure. The authors also examine the dependence of the solution on the Froude number in relation to Stokes waves.
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76E99
zbMATH DE Number: 1472263
zbMATH Keywords: sub-harmonic bifurcation
zbMATH Keywords: periodic function
zbMATH Keywords: steady periodic water waves
zbMATH Keywords: water of infinite depth
zbMATH Keywords: nonlinear pseudo-differential operator equation
zbMATH Keywords: non-local operator
zbMATH Keywords: Hilbert transform
zbMATH Keywords: arc-wise connected set of solutions
zbMATH Keywords: minimal period
zbMATH Keywords: first eigenvalue
zbMATH Keywords: linearized problem
zbMATH Keywords: local real analytic parametrization
zbMATH Keywords: Froude number
zbMATH Keywords: Stokes waves
author: Boris Buffoni
author: E. N. Dancer
author: John F. Toland
reviewed by: Alan Jeffrey
MaRDI profile type: Publication
full work available at URL: s002050000087
OpenAlex ID: W1979908744
DOI: 10.1007/S002050000087
description: scientific article; zbMATH DE number 1472263