an:01163510
Zbl 0907.58031
Magri, F.
Eight lectures on integrable systems. Written in collaboration with P. Casati, G. Falqui and M. Pedroni
EN
Kosmann-Schwarzbach, Yvette (ed.) et al., Integrability of nonlinear systems. Proceedings of the CIMPA school, Pondicherry Univ., India, January 8--26, 1996. Berlin: Springer. Lect. Notes Phys. 495, 256-296 (1997).
1997
a
37J35 37K10 35Q51 53D17
Gelfand-Dickey manifolds; integrable systems; Poisson manifolds; bihamiltonian manifolds; soliton equations; Casimir functions; KP equations; extended Lax representations; Poisson-Nijenhuis manifolds; Calogero systems
This is an introduction to the theory of integrable systems from the viewpoint of Poisson manifolds. This leads to bi-Hamiltonian manifolds and an elegant theory due to Casati, Falqui, Magri, and Pedroni. One produces Gelfand-Dickey (GD) manifolds via Marsden-Ratiu reduction from a class of Poisson manifolds relevant to the theory of soliton equations. These are phase spaces when the soliton equations are defined. Casimir functions and GD equations are developed and relations between Kadomtsev-Petviashvili (KP) and GD theories are indicated. KP equations arise as local conservation laws associated with GD equations. One also develops extended Lax representations and Poisson-Nijenhuis manifolds in connection with Calogero systems.
This work is quite fascinating.
For the entire collection see [Zbl 0879.00077].
Robert Carroll (Urbana)