Symmetry and the bifurcation of capillary-gravity waves
type: Item
instance of: scholarly article
title: Symmetry and the bifurcation of capillary-gravity waves
zbMATH Open document ID: 0615.76023
DOI: 10.1007/BF00251412
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: On considère les ondes de gravité-capillarité, irrotationnaires, stationnaires et périodiques à la surface d'un fluide parfait de profondeur infinie. Après la théorie faiblement non linéaire de \textit{B. Chen} et \textit{P. G. Saffman} [Stud. Appl. Math. 60, 183-210 (1979; Zbl 0429.76013)] les auteurs se proposent ici de donner un calcul mathématique rigoureux de l'existence de telles ondes de faible amplitude. Cette étude utilise la théorie de la bifurcation; elle est une conséquence de la procédure de réduction de Lyapunov-Schmidt. Elle est basée sur une formulation en termes d'équation intégrale d'un problème de frontière libre classique. Cette équation est linéarisée autour d'une fonction zéro correspondant à l'écoulement horizontal uniforme dont la vitesse joue le rôle de paramètre de la bifurcation. On étudie les bifurcations qui peuvent se produire lorsque ce paramètre varie, alors que la tension superficielle \(\gamma\) est fixée. Puis on étudie l'effet du changement de \(\gamma\) sur les diagrammes de bifurcation. Des valeurs critiques de \(\gamma\) sont signalées, celles pour lesquelles des bifurcations secondaires apparaissent ou disparaissent, par exemple.
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3997367
zbMATH Keywords: weakly nonlinear theory
zbMATH Keywords: steady periodic two-dimensional irrotational waves
zbMATH Keywords: perfect fluid
zbMATH Keywords: infinite depth
zbMATH Keywords: existence of waves
zbMATH Keywords: surface tension
zbMATH Keywords: capillary-gravity waves
zbMATH Keywords: steady, two-dimensional irrotational motion
zbMATH Keywords: bifurcation theory
zbMATH Keywords: symmetry
zbMATH Keywords: Lyapunov-Schmidt reduction procedure
author: Mark C. W. Jones
author: John F. Toland
MaRDI profile type: Publication
cites work: Q3768541
cites work: The existence of cnoidal water waves with surface tension
cites work: Applications of the blowing-up construction and algebraic geometry to bifurcation problems
cites work: Steady Gravity-Capillary Waves On Deep Water-1. Weakly Nonlinear Waves
cites work: Steady Gravity-Capillary Waves on Deep Water-II. Numerical Results for Finite Amplitude
cites work: Bifurcation from simple eigenvalues
cites work: Imperfect bifurcation in the presence of symmetry
cites work: Some effects of surface tension on steep water waves
cites work: Solitary and periodic gravity—capillary waves of finite amplitude
cites work: On Wilton ripples. II: Rigorous results
cites work: Boundary conditions and mode jumping in the buckling of a rectangular plate
cites work: Secondary Bifurcation Near a Double Eigenvalue
cites work: The bifurcation and secondary bifurcation of capillary-gravity waves
cites work: Q4039793
full work available at URL: https://doi.org/10.1007/bf00251412
OpenAlex ID: W2032144246
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description: scientific article; zbMATH DE number 3997367