Toda lattice hierarchy. I
type: Item
instance of: scholarly article
title: Toda lattice hierarchy. I
zbMATH Open document ID: 0552.35067
author: Kimio Ueno
author: Kanehisa Takasaki
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: A sequence of mutually commuting higher evolutions (i.e. a hierarchy) for the two dimensional Toda lattice equation (TL) \(\partial_{x_ 1}\partial_{y_ 1}u(s)=e^{u(s)-u(s-1)}-e^{u(s+1)-u(s)},\) with \(u(s)=u(s;x_ 1,y_ 1),\) \(s\in Z\), is presented. Let \(x=(x_ 1,x_ 2,...)\) and \(y=(y_ 1,y_ 2,...)\) be independent variables with infinitely many components. One introduces the matrices of size \(Z\times Z:\) \(L=\sum_{-\infty <n\leq 1}diag[b_ n(s;x,y)]\Lambda^ n,\quad b_ 1=0;\quad M=\sum_{-1\leq n<\infty}diag[c_ n(s;x,y)]\Lambda^ n,\quad c_{-1}\neq 0+\quad B_ n=(L^ n)_+=\sum^{n}_{m=0}diag[b_{n,m}(s;x,y)]\Lambda^{n- m};\quad C_ n=(M^ n)_-=\sum^{n- 1}_{m=0}diag[c_{n,m}(s;x,y)]\Lambda^{n-m}.\) The quantities \(b_ j(j\leq 1)\) and \(c_ j(j\geq -1)\) serve as unknown functions in the hierarchy. The TL hierarchy considered by the author is defined by the system of Lax type: \[ \partial_{x_ n}L=[B_ n,L],\quad \partial_{y_ n}L=[C_ n,L],\quad \partial_{x_ n}M=[B_ n,M],\quad \partial_{y_ n}M=[C_ n,M], \] n\(=1,2,... \). The linearization of the TL hierarchy is achieved by introducing the wave matrices \(W^{(\infty)}\) and \(W^{(0)}\) of size \(Z\times Z\) with the properties: \(LW^{(\infty)}=W^{(\infty)}\Lambda,\) \(MW^{(0)}=W^{(0)}\Lambda^{-1},\) \(\partial_{x_ n}W=B_ nW,\) \(\partial_{y_ n}W=C_ nW,\) \(W=W^{(\infty)},W^{(0)}.\) The bilinearization of the TL hierarchy is also obtained using the \(\sigma\) function defined by \(u(s)=\log [\sigma (s+1)/\sigma (s)].\)
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3879429
zbMATH Keywords: mutually commuting higher evolutions
zbMATH Keywords: Toda lattice
zbMATH Keywords: hierarchy
zbMATH Keywords: system of Lax type
zbMATH Keywords: linearization
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
full work available at URL: pjaa.59.167
OpenAlex ID: W2005689849
cites work: Q4745022
cites work: Two-dimensional generalized Toda lattice
DOI: 10.3792/PJAA.59.167
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description: scientific article; zbMATH DE number 3879429