Existence and orbital stability of cnoidal waves for a 1D Boussinesq equation
type: Item
instance of: scholarly article
title: Existence and orbital stability of cnoidal waves for a 1D Boussinesq equation
zbMATH Open document ID: 1146.35076
DOI: 10.1155/2007/52020
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/54532
review text: The authors of this very interesting paper study existence as well as nonlinear orbital stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equation written in the form \[ \Phi_{tt}-\Phi_{xx}+a\Phi_{xxxx}-b\Phi_{xxtt}+\Phi_{t}\Phi_{xx}+2\Phi_{x}\Phi_{xt}=0, \] where \(a\), \(b\) are positive numbers. This equation is a rescaled version of the one-dimensional Benney-Luke equation \[ \Phi_{tt}-\Phi_{xx}+\mu (a\Phi_{xxxx}-b\Phi_{xxtt})+\varepsilon (\Phi_{t}\Phi_{xx}+2\Phi_{x}\Phi_{xt})=0, \] which is derived from evolution of two-dimensional long water waves with surfaces tension. The function \(\Phi (x,t)\) is the nondimensional velocity potential at the bottom fluid boundary, \(\mu \) represents the long-wave parameter (dispersion coefficient), \(\varepsilon \) represents the amplitude parameter, and \(a-b+1/3=\sigma \) is the Bond number associated with the surface tension. Periodic travelling-wave solutions with an arbitrary fundamental period \(T_0\) are obtained by using Jacobian elliptic functions. It turns out that there exists stable (in orbital sense) solutions, that is a consequence of the general stability criteria established by \textit{M. Grillakis}, \textit{J. Shatah}, and \textit{W. Strauss} [J. Funct. Anal. 74, 160--197 (1987; Zbl 0656.35122)]. The periodic eigenvalue problem associated to the Lamé equation is studied as well.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K45
zbMATH DE Number: 5282464
zbMATH Keywords: orbital stability
zbMATH Keywords: cnoidal waves
zbMATH Keywords: Boussinesq equation
zbMATH Keywords: travelling waves
zbMATH Keywords: Benney-Luke equation
author: Jaime Angulo Pava
author: José R. Quintero
MaRDI profile type: Publication
OpenAlex ID: W2070669548
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description: scientific article; zbMATH DE number 5282464