Integration of non-Abelian Langmuir type lattices by the inverse spectral problem method
type: Item
instance of: scholarly article
title: Integration of non-Abelian Langmuir type lattices by the inverse spectral problem method
zbMATH Open document ID: 0907.34047
DOI: 10.1007/BF02466009
author: A. S. Osipov
publication date: 1998-10-11T00:00:00Z
review text: The author deals with Cauchy problems of the type \[ \dot C_n= \sum^q_{i= 1} C_{n+ i}C_n- C_n \sum^q_{i= 1} C_{n- i}\quad (q\in\mathbb{N}),\tag{1} \] \[ C_n= C_n(t)\in{\mathcal L}(B),\quad t\in [0,T],\quad 0<T\leq\infty,\quad \cdot={d\over dt}, \] in semi-infinite \((n\in\mathbb{Z}_+)\) and infinite \((n\in\mathbb{Z})\) cases; here \(B\) is a Banach space. In the scale case for \(q= 1\) and positive \(C_n\) this system is the well-known Langmuir chain, for \(q> 1\) it is an discrete analogue of the Korteweg-de Vries equation. The author describes an effective procedure for constructing bounded solutions to (1) in terms of the corresponding Lax representation \(\dot L= [L,A]\) of system (1) and its Weil matrix \({\mathfrak M}(\lambda)\). The main results are necessary and sufficient conditions (of algebraic type) under which this procedure really defines a unique solution to (1).
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 81U40
zbMATH DE Number: 1179571
zbMATH Keywords: Cauchy problems
zbMATH Keywords: Langmuir chain
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: bounded solutions
zbMATH Keywords: Lax representation
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
cites work: On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices
cites work: The integration of semi-infinite Toda chain by means of inverse spectral problem
cites work: Highest weight representations of the Lie algebra \(\mathfrak{gl}_{\infty}\)
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 1179571