Cauchy problem for modified Korteweg-de Vries equation with piecewise- type initial data
type: Item
instance of: scholarly article
title: Cauchy problem for modified Korteweg-de Vries equation with piecewise- type initial data
zbMATH Open document ID: 0567.35074
DOI: 10.1007/BF00968683
author: E. P. Zhidkov
author: Kiril Kirchev
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The modified Korteweg-de Vries (mKdV) equation \(u_ t-u^ 2u_ x+u_{xxx}=0\) with \(u(x,0)=f(x)\), \(x\in R\), \(t\geq 0\) is considered where the initial data f(x) belong to a certain Banach space \(X^ s\), \(s\geq 3\). By the use of the method of pseudoparabolic regularization the existence, uniqueness and continuous dependence on initial data of the solution of the Cauchy problem for the mKdV-equation is proved. Stability of the form of the solitary wave solution belonging to \(X^ s\) at any t is proved. The existence and uniqueness of the solution of the Cauchy problem for the Benjamin-Bona-Mahony equation \(u_ t+u_ x+u^ 2u_ x-u_{xxt}=0\) is also proved.
reviewed by: B. Konopelchenko
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 3905118
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: initial data
zbMATH Keywords: Banach space
zbMATH Keywords: pseudoparabolic regularization
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: continuous dependence
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Stability
zbMATH Keywords: solitary wave solution
zbMATH Keywords: Benjamin-Bona-Mahony equation
MaRDI profile type: Publication
cites work: Method for Solving the Korteweg-deVries Equation
cites work: Q4098096
cites work: Q4177990
cites work: The initial-value problem for the Korteweg-de Vries equation
cites work: Model equations for long waves in nonlinear dispersive systems
cites work: Q3674888
cites work: Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
cites work: Q3279399
description: scientific article; zbMATH DE number 3905118