Poisson-Nijenhuis Structures and Sato Hierarchy

The thesis is organized in three distinct parts. The first part ( § 1 and §2) is purely expository. After a brief introduction to the basic ideas of the bi-Hamiltonian approach to integrable systems in §1, in §2 the bi-Hamiltonian factorization of Sato's equations is described as explicitly as possible. This section collects the "experimental facts" which we aim to explain in the following part of the thesis. In §2 we also introduce and discuss different representations of the Sato hierarchy, as families of integro-differential equations in two space variables and in a finite number of fields, which we call Sato-Gel'fand-Dikii hierarchies. The simplest representation of this type coincides with the well-known KP hierarchy, while the other representations do not appear, to oμr knowledge, in the previous literature. The second part of the thesis includes §3, §4 and §5 and represents the theoretical core of this work. The main result (§4) is the construction of the Poisson-Nijenhuis structures already mentioned; in §5 we introduce the Kac-Moody algebra of Hamiltonian vector fields, and we show that these vector fields admit a Lax representation. The only arbitrary point in the construction is the choice of a Lie-algebra cocycle corresponding to the affine part of the Lie-Poisson brackets: both the Kac-Moody algebra of bi-Hamiltonian vector fields and its Lax representation are completely determined by that cocycle. The third and final part is devoted to the formal application of the abstract framework to algebras of pseudodifferential operators. In §6 the Gel'fand-Dikii and the Sato-Gel'fandDikii hierarchies are obtained as reductions (on different affine subspaces) of the dynamical systems previously obtained, for finite n, while the Sato hierarchy is recovered in §7 as a generalization of the same construction to the case n = oo.