Asymptotic representation of surface waves in the form of two traveling Burgers waves
type: Item
instance of: scholarly article
title: Asymptotic representation of surface waves in the form of two traveling Burgers waves
zbMATH Open document ID: 0857.35146
DOI: 10.1007/BF01077050
author: Pavel I. Naumkin
author: I. A. Shishmarev
publication date: 1997-03-09T00:00:00Z
review text: The authors study the Cauchy problem for the system \[ \psi_t+(\psi\varphi)_x+ K_{11}\psi+K_{12}\varphi=0, \qquad \varphi_t+0,5(\varphi^2)_x+ K_{21}\psi+K_{22}\varphi=0, \] where \(\psi\) and \(\varphi\) are real functions, \(x\in \mathbb{R}^1\), \(t>0\), and \(K_{ij}\) are linear pseudodifferential operators defined via the inverse Fourier transform. This system covers numerous well-known systems from mechanics; in particular, a special choice of operators \(K_{ij}\) provides a system describing surface water waves with allowance for viscosity and surface tension. Under some assumptions on the eigenvalues of the matrix \((K_{ij})\) it is proved that for \(t\to \infty\) uniformly with respect to \(x\) \[ \psi(x,t)= b\nu_1(x,t)+ b\nu_2(x,t)+ O(t^{-0,5-\gamma}), \qquad \varphi(x,t)= -\nu_1(x,t)+ \nu_2(x,t)+ O(t^{-0,5-\gamma}), \] where \(b\) and \(\gamma>0\) are constants and \(\nu_i(x,t)\) are solutions of the Cauchy problem for the system of Burgers' equations with transfer.
Mathematics Subject Classification ID: 35S10
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 898972
zbMATH Keywords: Cauchy problem
zbMATH Keywords: linear pseudodifferential operators
zbMATH Keywords: inverse Fourier transform
zbMATH Keywords: eigenvalues
zbMATH Keywords: Burgers equations with transfer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01077050
OpenAlex ID: W1994265728
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 898972