Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 034, 23 pages      arXiv:0803.3866      http://dx.doi.org/10.3842/SIGMA.2008.034
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

Gloria Marí Beffa
Department of Mathematics, University of Wisconsin, Madison, WI 53705, USA

Received November 14, 2007, in final form March 13, 2008; Published online March 27, 2008

Abstract
In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection.

Key words: invariant evolutions of curves; Hermitian symmetric spaces; Poisson brackets; differential invariants; projective differential invariants; equations of KdV type; completely integrable PDEs; moving frames; geometric realizations.

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