Bäcklund transformations and loop group actions

We construct a local action of the group of rational maps from 𝕊2 to GL(n, ℂ), on local solutions of flows of the ZS-AKNS sl(n, ℂ)-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different factorizations of a quadratic element. We prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities. However, the action of the subgroup of the rational maps that satisfy the U(n)-reality condition on the space of global rapidly decaying solutions of the flows in the u(n)-hierarchy is global, and the action of a simple element gives a global Bäcklund transformation. The actions of certain elements in the rational loop group on the vacuum give rise to explicit time-periodic multisolitons (multibreathers). We show that this theory generalizes the classical Bäcklund theory of the sine-Gordon equation. The group structures of Bäcklund transformations for various hierarchies are determined by their reality conditions. We identify the reality conditions (the group structures) for the sl(n, ℝ), u(k,n − k), KdV, Kupershmidt-Wilson, and Gel'fand-Dikii hierarchies. The actions of linear fractional transformations that satisfy a reality condition, modulo the center of the group of rational maps, give Bäcklund and Darboux transformations for the hierarchy defined by the reality condition. Since the factorization cannot always be carried out under this reality condition, the action is again local, and Bäcklund transformations only generate local solutions for these hierarchies unless singular solutions are allowed.