Poisson actions and scattering theory for integrable systems. (English) Zbl 0935.35163

Terng, Chuu-Lian; Uhlenbeck, Karen

Terng, Chuu Lian (ed.) et al., Surveys in differential geometry. Vol. IV. A supplement to the Journal of Differential Geometry. Integral systems (integrable systems). Lectures on geometry and topology. Cambridge, MA: International Press. 315-402 (1998).

One identifies conservation laws, hierarchies, scattering theory, and Bäcklund transformations as facets of a theory of Poisson group actions and applies the theory to the ZS-AKNS \(n\times n\) matrix hierarchy. Topics include Poisson actions, negative flows in the decay case, Poisson structure for negative flows (decay case), action of the rational loop group, scattering data and Birkhoff decompositions, Poisson structure for the positive flows (asymptotic case), symplectic structures for the restricted case, Bäcklund transformations for the \(j\)th flow, geometric nonlinear Schrödinger equation, and first flows and flat metrics.
For the entire collection see [Zbl 0918.00013].

Reviewer: Robert Carroll (Urbana)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35P25	Scattering theory for PDEs
37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures

Keywords:

conservation laws; scattering theory; Bäcklund transformations; Poisson group actions; ZS-AKNS \(n\times n\) matrix hierarchy; negative flows; rational loop group; Birkhoff decompositions; Poisson structure; positive flows; symplectic structures