Integrable systems as nonlinear realizations of infinite-dimensional symmetries: The Liouville equation example
type: Item
instance of: scholarly article
title: Integrable systems as nonlinear realizations of infinite-dimensional symmetries: The Liouville equation example
zbMATH Open document ID: 0541.35075
DOI: 10.1007/BF00420039
author: S. Krivonos
author: Evgeny Ivanov
published in: Letters in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The Liouville equation \(U_{+-}=m^ 2\exp(-2u)\) is shown to have an adequate description in terms of the nonlinear realization of infinite- parameter conformal group G in \((1+1)\)-dimensions. The \((1+1)\)-Minkowski space coordinates \(x^{\pm},x^ =\) and the field u(x) are identified with certain parameters of the coset G/H, H being the \((1+1)\)-Lorentz group. The Liouville equation appears as one of the covariant constraints reducing this coset space to its connected geodesic subspace \(SL(2,R)/H.\) An alternative reduction to the subspace \({\mathfrak P}(1,1)/H\) (\({\mathfrak P}(1,1)\) is the \((1+1)\)-Poincaré group), yields the free equation for u(x). The corresponding zero-curvature representations naturally arise as the consequences of the Maurer-Cartan equations for G/H and the covariant reduction conditions. Bäcklund transformations also obtain a simple group-theoretical interpretation within this approach: these are represented by the constrained right gauge shifts on the coset G/H. A possibility of extending the method to other integrable system is discussed.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 3859642
zbMATH Keywords: Liouville equation
zbMATH Keywords: nonlinear realization
zbMATH Keywords: Maurer-Cartan equations
zbMATH Keywords: Bäcklund transformations
zbMATH Keywords: gauge shifts
MaRDI profile type: Publication
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cites work: Q3265258
cites work: Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws.
full work available at URL: https://doi.org/10.1007/bf00420039
OpenAlex ID: W2010763124
description: scientific article; zbMATH DE number 3859642