On solitary waves and nonlinear oscillations in a stratified fluid with surface tension
type: Item
instance of: scholarly article
title: On solitary waves and nonlinear oscillations in a stratified fluid with surface tension
zbMATH Open document ID: 0814.76084
author: V. Pereyra
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-08-22T00:00:00Z
review text: The authors consider nonlinear two-dimensional flows in an ideal stratified liquid. It was previously known that in the case when the linearized equations for this system had not more than two positive eigenvalues, there existed a solution to the full system of nonlinear governing equations in the form of a solitary wave on top of an oscillatory part; the main technical problem was to single out the oscillatory part of the solution. A crucial point in the previous work was the fact that the number of available arbitrary constants in the solution was just sufficient to satisfy all the necessary resolvability conditions. When one has more than three positive eigenvalues in the spectrum of the linearized problem, this is not, generally speaking, possible. However, the existence proof for the solution in the form of the solitary wave on top of oscillations can be extended to this case if the stationary state is subject to a certain condition. This is the subject of the present work. The proof is then extended to the case of an arbitrary number of the positive eigenvalues in the linear spectrum.
Mathematics Subject Classification ID: 76V05
Mathematics Subject Classification ID: 76B25
Mathematics Subject Classification ID: 76B45
zbMATH DE Number: 599916
zbMATH Keywords: positive eigenvalues
zbMATH Keywords: resolvability conditions
zbMATH Keywords: linearized problem
zbMATH Keywords: existence
zbMATH Keywords: linear spectrum
reviewed by: B. A. Malomed
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1162
OpenAlex ID: W2082204819
DOI: 10.1006/JMAA.1994.1162
description: scientific article; zbMATH DE number 599916