Poisson Actions and Scattering Theory for Integrable Systems

Abstract

Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger equation, modified KdV, and the n-wave equation). We also discuss a number of applications in geometry, including the sine-Gordon equation, harmonic maps, Schrödinger flows on Hermitian symmetric spaces, Darboux orthogonal coordinates, and isometric immerisons of one space form in another.

85 pages

References in corpus (1)

• Bäcklund Transformations and Loop Group Actions

Cited by in corpus (5)

• New Hamiltonian formalism and Lagrangian representations for integrable hydrodynamic type systems
• Transformations of flat Lagrangian immersions and Egoroff nets
• Multiply conjugate systems containing deformations of quadrics
• The Method of Archimedes in the geometry of quadrics
• Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies - An Analytic Approach