Gauge theory of Bäcklund transformations. II
type: Item
instance of: scholarly article
title: Gauge theory of Bäcklund transformations. II
zbMATH Open document ID: 0658.35080
DOI: 10.1016/0167-2789(87)90227-2
author: V. D. Zurkowski
author: David H. Sattinger
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: [For part I, see Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, 273-300 (1987; Zbl 0643.35092).] A class of gauge transformations is constructed for Hamiltonian hierarchies of completely integrable systems on semi-simple Lie algebras. These transformations create or annihilate poles in the scattering data, hence create or annihilate solitons in the potential Q. In that sense they generalize Bäcklund transformations to which they reduce precisely for the \(2\times 2\) ``AKNS'' systems. To each Bäcklund-gauge transformation is associated an automorphism in the Weyl group of the Lie algebra. Two proofs are given to show that these Bäcklund-gauge transformations preserve the nonlinear evolution equation. One proof uses the transformation of the scattering data under the Bäcklund transformation; the second proof is purely gauge theoretic.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 58J72
Mathematics Subject Classification ID: 58J90
zbMATH DE Number: 4075679
zbMATH Keywords: gauge transformations
zbMATH Keywords: Hamiltonian hierarchies
zbMATH Keywords: completely integrable systems on semi-simple Lie algebras
zbMATH Keywords: scattering data
zbMATH Keywords: solitons
zbMATH Keywords: Bäcklund transformations
zbMATH Keywords: Bäcklund-gauge transformation
zbMATH Keywords: nonlinear evolution equation
MaRDI profile type: Publication
full work available at URL: 0167-2789(87)90227-2
OpenAlex ID: W2474546649
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description: scientific article; zbMATH DE number 4075679