Gauge theory of Bäcklund transformations. II
[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.
completely integrable systems on semi-simple Lie algebras
scientific article; zbMATH DE number 4075679
scientific article