Pseudo-Differential Operators and Integrable Models

The importance of the theory of pseudo-differential operators in the study of non linear integrable systems is point out. Principally, the algebra $\Xi$ of nonlinear (local and nonlocal) 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 subalgebras $\Sigma_{jr}, j=0,\pm 1, r=\pm 1$ completely specified by the quantum numbers: $s$ and $(p,q)$ describing respectively the conformal weight (or spin) and the lowest and highest degrees. The algebra ${\huge \Sigma}_{++}$ (and its dual $\Sigma_{--}$) of local (pure nonlocal) differential operators is important in the sense that it gives rise to the explicit form of the second hamiltonian structure of the KdV system and that we call also the Gelfand-Dickey Poisson bracket. This is explicitly done in several previous studies, see for the moment \cite{4, 5, 14}. Some results concerning the KdV and Boussinesq hierarchies are derived explicitly.