Difficulties with SDiff(2) Toda equation
type: Item
instance of: scholarly article
zbMATH Open document ID: 1005.37046
publication date: 2002-06-23T00:00:00Z
Mathematics Subject Classification ID: 37K60
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 37K30
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 35Q75
zbMATH DE Number: 1721631
zbMATH Keywords: Lax pairs
zbMATH Keywords: Bäcklund transformation
zbMATH Keywords: Toda lattice
author: J. D. III Finley
MaRDI profile type: Publication
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title: Difficulties with SDiff(2) Toda equation
review text: This paper is devoted to the SDiff(2) Toda equation, which can be written for one unknown function of three variables, with differentiation shown by a subscript: NEWLINE\[NEWLINEu_{,z\overline{z}}= e^{u_{,tt}} \iff v_{,z\overline{z}}= (e^v)_{,tt}, \qquad v\equiv u_{,tt}. \tag{1}NEWLINE\]NEWLINE It is commonly accepted that to find a more direct method to ``buy'' new solutions from old ones begins with the Bäcklund transformation for the (2-dimensional) Toda lattice built over the Cartan matrix for the Lie algebra \(A_n\) or \(A_n^{(1)}\), and consider the limit as \(n\to\infty\). While this limit gives the derived PDEs, the associated limit of the (first-order) PDEs that define the Bäcklund transformations gives only trivial results. The author analyzes this phenomena as well as well-known ``Lax pairs''.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].
reviewed by: Messoud Efendiev
description: scientific article; zbMATH DE number 1721631