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𝒟 -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

We study the geometry of 𝒟 -bundles—locally projective 𝒟 -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of 𝒟 -bundles; in particular, we prove that the local structure of 𝒟 -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...

A differential geometric setting for dynamic equivalence and dynamic linearization

Jean-Baptiste Pomet (1995)

Banach Center Publications

This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise: equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms. These diffeomorphisms are very much related to Lie-Bäcklund transformations. It is proved in this framework that dynamic equivalence of single-input systems is the same as static equivalence.

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