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Bäcklund transformations and loop group actions. (English) Zbl 1031.37064
The article describes Bäcklund transformations of integrable systems as a special case of dressing actions. The general dressing action of integrable systems, introduced by Zakharov and Shabat, is a local action of the full loop group defined in terms of loop group factorization. According to the article under review, the special case of Bäcklund transformations corresponds to the action of fractional linear loops, the so called simple elements. In this setting, Bianchi permutability theorems are obtained by applying different factorizations of quadratic elements into simple elements.
While in the general case of dressing actions an explicit computation of the loop group factorization amounts to solving a Riemann-Hilbert problem, the computation of Bäcklund transformations is purely algebraic, provided one already has a trivialization of the flat connection corresponding to the Lax pair. In particular, a new solution of a nonlinear partial differential equation (the compatibility or zero-curvature equation) can be computed from a given one by solving ordinary differential equations (namely those corresponding to trivializing the underlying flat connection).
The focus of the article is on the ZS-AKNS $$sl(n,\mathbb{C})$$-hierarchy, whose different real forms include the famous KdV-, mKdV- and NLS-equations as well as the sine-Gordon equation. It is shown that the classical Bäcklund transformation of the sine-Gordon equation is a special case of the general Bäcklund transformation.

##### MSC:
 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 35Q53 KdV equations (Korteweg-de Vries equations) 17B80 Applications of Lie algebras and superalgebras to integrable systems 22E67 Loop groups and related constructions, group-theoretic treatment 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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