The reductive perturbation method and the Korteweg-de Vries hierarchy
type: Item
instance of: scholarly article
title: The reductive perturbation method and the Korteweg-de Vries hierarchy
zbMATH Open document ID: 0841.76010
DOI: 10.1007/BF00994645
author: M. A. Manna
author: Roberto A. Kraenkel
author: J. G. Pereira
published in: Acta Applicandae Mathematicae
publication date: 1995-08-27T00:00:00Z
review text: The authors consider a two-dimensional inviscid incompressible fluid in a constant gravitational field. The space coordinates are denoted by \((x,z)\), and the corresponding components of the velocity \(\vec v\) by \((u,w)\). The gravitational acceleration \(g\) is in the negative \(z\) direction. If the fluid is irrotational, the equations describing it are \(\nabla^2 \varphi=0\) and \((p- p_0) /\rho=- {{\partial \varphi} \over {\partial t}}- {1\over 2} (\nabla \varphi)^2- gz\), where \(\rho\) is the fluid density, \(p\) is the pressure, \(\varphi\) is the velocity potential, and \(p_0\) is an integration constant. By using a reductive perturbation method with the introduction of an infinite sequence of slow time variables \(\tau_1, \tau_2, \tau_3, \dots\), the propagation of long surface waves in a shallow inviscid fluid is studied. The Korteweg-de Vries equation appears as the lowest order amplitude equation in slow variables. The paper is well written and organized.
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76B25
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 790809
zbMATH Keywords: multiple time formalism
zbMATH Keywords: higher-order evolution equations
zbMATH Keywords: constant gravitational fluid
zbMATH Keywords: velocity potential
zbMATH Keywords: shallow fluid
zbMATH Keywords: long surface waves
reviewed by: A. Klíč
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00994645
OpenAlex ID: W4255779460
description: scientific article; zbMATH DE number 790809