A model system for strong interaction between internal solitary waves
type: Item
instance of: scholarly article
title: A model system for strong interaction between internal solitary waves
zbMATH Open document ID: 0752.35056
DOI: 10.1007/BF02099010
author: G. Ponce
author: Jerry L. Bona
author: Jean-Claude Saut
author: Michael Mudi Tom
published in: Communications in Mathematical Physics
publication date: 1992-09-26T00:00:00Z
review text: The model for the strong interaction of long internal gravity waves propagating on neighboring pycnoclines in a stratified fluid is described by the system of the equations [\textit{J. A. Gear} and \textit{R. Grimshow}, Stud. Appl. Math. 70, 235-258 (1984; Zbl 0548.76020)], \[ u_ t+uu_ x+u_{xxx}+a_ 3v_{xxx}+a_ 1vv_ x+a_ 2(uv)_ x=0,\tag{1} \] \[ b_ 1v_ t+rv_ x+vv_ x+v_{xxx}+b_ 2a_ 3u_{xxx}+b_ 2a_ 2uu_ x+b_ 2a_ 1(uv)_ x=0, \tag{2} \] where \(a_ 1\), \(a_ 2\), \(a_ 3\), \(b_ 1>0\), \(b_ 2>0\), and \(r\) are real constants. The Cauchy problem for the system (1), (2) is shown to be globally well-posed in suitably strong function spaces. The abstract evolution equations together with somewhat delicate estimates are obtained using techniques from harmonic analysis. The system (1), (2) is shown to be susceptible to the blow-up phenomenon.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 76B55
zbMATH DE Number: 54714
zbMATH Keywords: coupled KdV-type equations
zbMATH Keywords: blow-up solutions
zbMATH Keywords: neighboring pycnoclines
zbMATH Keywords: stratified fluid
zbMATH Keywords: Cauchy problem
zbMATH Keywords: abstract evolution equations
zbMATH Keywords: estimates
zbMATH Keywords: harmonic analysis
reviewed by: Yuri S. Kivshar
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 54714