Pseudo-differential Operators and Integrable Models

The importance or the theory or pseudo-differential operators In the study or non-linear integrable systems is pointed out. Principally, the algebra Xi of non-linear (local and non-local) differential operators. acting on the ring of analytic functions u(s)(x, t), is studied. It is shown, In particular, that this space splits into several Classes of sunahigebras Sigma(jr), j = = 0, +/- 1, r = +/- 1 completely specified by the quantum numbers s and (p, q) describing the conformal weight (or spin) and the lowest and highest degrees, respectively. The algebra Sigma(++) (and its dual Sigma(--)) or local (pure non-local) differential operators is important in the sense that it gives rise to an explicit form of the second Hamiltonian structure of the Kdv system that we call the Cielfarid-flickAay Poleson bracket. This is explicitly done in several previous studies, see [4, 5, 14]. Some results concerning the KcIV and Boussinetin hierarchies are derived explicitly.