Covariant differential operators for densities and KdV equation as a flow on diffeomorphism groups
type: Item
instance of: scholarly article
title: Covariant differential operators for densities and KdV equation as a flow on diffeomorphism groups
zbMATH Open document ID: 0685.47028
DOI: 10.1007/BF00420013
author: W. Scherer
published in: Letters in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: In the main statements of this paper the author has presented an algorithm for constructing differential operators of the form \[ D_{\hat q}^{(k)}=(d/d\theta)^ k+\sum^{k-2}_{j=0}A^ k_ j(q)(\theta)(d/d\theta \quad)^ j,\quad k=2,3,4,..., \] such that \(D_{\hat q}^{(k)}: \Omega_{(1-k)/2}(S^ 1)\to \Omega_{(1+k)/2}(S^ 1),\) where \(\Omega_ u(S^ 1)\) is the space of densities of weight u, and \(D_{\hat q}^{(k)}\) has properties similar to those of Hill's operator \(D_{\hat q}^{(2)}=(d/d\theta)^ 2+q(\theta).\) The introduction of the paper contains an indication of the connections of the equations with quantization in string theory, and further considerations indicate connections with the KdV equation \(\dot q_ t=q_ t^{(3)}+3q_ tq'_ t,\) which occurs as a necessary condition for the existence of a solution of \[ D_{\hat q}^{(2)}\psi_ t=0;\quad {\dot \psi}_ t=-v'_ t\psi_ t+v_ t\psi '_ t. \]
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 81T30
zbMATH DE Number: 4122674
zbMATH Keywords: Hill's operator
zbMATH Keywords: quantization in string theory
zbMATH Keywords: KdV equation
reviewed by: George O. Okikiolu
MaRDI profile type: Publication
Wikidata QID: Q115395004
cites work: Orbits of the group of diffeomorphisms of a circle and local Lie superalgebras
cites work: Integrals of nonlinear equations of evolution and solitary waves
cites work: Nonlinear-evolution equations of physical significance
cites work: Holomorphic structure of superstring vacua
description: scientific article; zbMATH DE number 4122674