On the transverse instability of one dimensional capillary-gravity waves
type: Item
instance of: scholarly article
title: On the transverse instability of one dimensional capillary-gravity waves
zbMATH Open document ID: 1197.35333
author: Frédéric Rousset
author: Nikolay Tzvetkov
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2010-06-02T00:00:00Z
review text: The aim of this article is to prove the Lyapounov orbital instability of the line solitary water waves constructed by \textit{C. J. Amick} and \textit{K. Kirchgässner} [Arch. Ration. Mech. Analysis, 105, 1--49 (1989; Zbl 0666.76046)] governed by the system \[ \eta_{t}=\eta_{x}+G[\eta]\varphi, \] \[ \varphi_{t}=\varphi_{x}-\frac{1}{2}|\nabla\varphi|^{2}+\frac{1}{2}\frac{(G[\eta]\varphi+\nabla\varphi\cdot\nabla\eta)^{2}}{1+|\nabla\eta|^{2}}-\alpha\eta+\beta\nabla\cdot\frac{\nabla\eta}{\sqrt{1+|\nabla\eta|^{2}}} . \] Here \(\eta=\eta(t,x,y)\), \(\varphi=\varphi(t,x,y)\), \(t,x,y\in\mathbb{R}\), \(\nabla=(\partial_{x},\partial_{y})\) and \(\alpha=\frac{gh}{c^{2}}\), \(\beta=\frac{b}{hc^{2}},\) \(c\) represents the propagation speed, \(g\) the gravity constant, \(b\) takes into account the surface tension effects, \(h\) represents the deepness of the fluid domains and \(G[\eta]\) is a Dirichlet-Neumann map to be defined shortly with respect to transverse perturbations.
reviewed by: Svetlana A. Grishina
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35C08
Mathematics Subject Classification ID: 35Q31
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76B45
zbMATH DE Number: 5715579
zbMATH Keywords: free boundary problems
zbMATH Keywords: stability
zbMATH Keywords: solitary waves
zbMATH Keywords: water-waves
MaRDI profile type: Publication
full work available at URL: dcdsb.2010.13.859
OpenAlex ID: W1987198037
DOI: 10.3934/DCDSB.2010.13.859
description: scientific article; zbMATH DE number 5715579