Model equation of surface waves of viscous fluid down an inclined plane
type: Item
instance of: scholarly article
title: Model equation of surface waves of viscous fluid down an inclined plane
zbMATH Open document ID: 0855.35116
author: Htay Aung Win
published in: Journal of Mathematics of Kyoto University
publication date: 1997-02-03T00:00:00Z
review text: The purpose of this paper is to study the periodic solutions of the following partial differential equation (we call it Korteweg-de Vries-Kuramoto-Sivashinsky (KdV-KS) equation): \[ u_t+ u_{xxxx}+ u_{xxx}+ Ru_{xx}+ uu_x= 0,\tag{1} \] where \(R\) is the parameter corresponding to the Reynolds number. It is an approximate equation for surface waves of a two-dimensional incompressible viscous fluid down an inclined plane under the assumption of small amplitude and long wave. We investigate the form of time periodic solutions of (1) bifurcating from the zero solution. We show that (1) has a Hopf bifurcation from the zero solution, however KS equation does not. Moreover, we consider the travelling wave solution of KdV-KS equation \(u(z)= u(x- ct)\), and prove that it has periodic solutions bifurcating from the zero solution. Additionally, we prove that the time periodic solution bifurcating from the zero solution corresponds to some travelling wave solution.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 76B15
zbMATH DE Number: 567079
zbMATH Keywords: Korteweg-de Vries-Kuramoto-Sivashinsky equation
zbMATH Keywords: surface waves
zbMATH Keywords: Hopf bifurcation from the zero solution
zbMATH Keywords: travelling wave solution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519194
OpenAlex ID: W1542370041
DOI: 10.1215/KJM/1250519194
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description: scientific article; zbMATH DE number 567079