Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

Abstract

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.

On an integrable multi-component Camassa–Holm system arising from Möbius geometry

“…A variety of integrable equations including the NLS equation [32], the KdV equation [33], the mKdV equation [34], the Sawada-Kotera equation [6,35] and the Kaup-Kupershmidt equation [6,36,37] and certain multi-component generalizations have explicit geometric realizations. Moreover, Miura transformations, Bäcklund transformations and bi-Hamiltonian structures [16,19] of integrable systems have nice geometric formulations (see [38][39][40][41][42][43][44][45] and references therein). Indeed, the Hamiltonian structures can be generated by the classical Possion brackets defined on spaces of loops on duals of Lie algebra via their reduction to the space of Maurer-Cartan invariants.…”

Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

“…On the other hand, it is known that a lot of completely integrable systems are described as bi-Hamiltonian systems, from which the existence of many first integrals can be deduced (Magri's theorem [22,27]). In this context, many of motions of curves as above have been studied from the viewpoint of bi-Hamiltonian systems recently [1,2,3,4,5,6,7,8,21,23,24,31]. The purpose of this paper is to construct a multi-Hamiltonian structure associated to the higher KdV flows on each level set of Hamiltonian functions in a geometric way (Theorem 7).…”

Quaternionic soliton equations from Hamiltonian curve flows in {\bb H}{\bb P}^n

“…Certain geometric flows of curves in homogeneous plane and space geometries are well known to encode scalar soliton equations through the induced evolution of geometrical invariants of the curve [1][2][3][4][5][6][7][8][9][10]. There has been much recent interest in extending such geometrical derivations to multi-component soliton equations.…”