Canonical forms for bihamiltonian systems. (English) Zbl 0823.58019

Olver, Peter J.

Babelon, Olivier (ed.) et al., Integrable systems: the Verdier memorial conference. Actes du colloque international de Luminy, France, July 1-5, 1991. Boston, MA: Birkhäuser. Prog. Math. 115, 239-249 (1993).

A system of differential equations is called bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways: \(dx/dt = J_ 1\nabla H_ 1= J_ 2\nabla H_ 0\). The bi-Hamiltonian structure determined by \(J_ 1\), \(J_ 2\), is compatible if the sum \(J_ 2+ J_ 2\) is also Hamiltonian. The bi-Hamiltonian structure is nondegenerate if the operator \(J_ 1\) is nonsingular.
First the author reviews some results on the canonical forms for compatible, nondegenerate complex-analytic bi-Hamiltonian systems. Secondly, the author briefly describes the results of Brouzet and Fernandes on integrable Hamiltonian systems without bi-Hamiltonian structures.
For the entire collection see [Zbl 0807.00017].

Reviewer: Li Jibin (Kunming)

MSC:

37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51	Soliton equations

Keywords:

integrability; canonical forms; bi-Hamiltonian systems