The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition
type: Item
instance of: scholarly article
title: The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition
zbMATH Open document ID: 1206.37033
published in: Siberian Mathematical Journal
publication date: 2010-11-15T00:00:00Z
review text: The author studied the following problem for the Volterra chain: \[ \dot{a}_{n}=\tfrac{1}{2}a_{n}(a_{n-1}^{2}-a_{n+1}^{2}), \quad n=1,2,\dots \tag{1} \] \[ a_{n}(0)-\hat{a}\to 0, \quad\text{as}\quad n \to\infty, \tag{2} \] \[ a_{0}=0, \tag{3} \] where \(\hat{a}_{n+N}=\hat{a}_{n}>0\), \(n=1,2,\dots,N \) is a positive integer, and the dot above a symbol stands for derivative with respect to \( t\). Looking for solution \( a_{n}(t)\) such that \(a_{n}(0)-\hat{a}\) is rapidly decreasing, i.e. for every \(T>0\) \[ \bigg\|\sum^\infty_{n=1} n|a_n(t)-\hat{a}|\bigg\|_{C[0,T]}<\infty. \tag{4} \] Justified the method of inverse scattering problem and thereby the global solvability of (1--3) in (4) for the initial data with finite third moment, also provse a theorem about the absence of solutions.
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 34K29
zbMATH DE Number: 5816249
zbMATH Keywords: Volterra chain
zbMATH Keywords: scattering problem
zbMATH Keywords: inverse problem method
zbMATH Keywords: Toda chain
author: Agil Kh. Khanmamedov
MaRDI profile type: Publication
full work available at URL: s11202-010-0036-1
OpenAlex ID: W2123438719
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DOI: 10.1007/S11202-010-0036-1
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description: scientific article; zbMATH DE number 5816249