Pseudo-Differential Operators and Integrable Models

TL;DR

This paper explores the algebraic structure of pseudo-differential operators and their role in integrable systems like KdV, providing explicit results on their subalgebras and associated Hamiltonian structures.

Contribution

It introduces a detailed classification of the algebra of differential operators based on quantum numbers and links these structures to integrable models such as KdV.

Findings

Algebra splits into subalgebras specified by quantum numbers.

Local and nonlocal operators relate to Hamiltonian structures of KdV.

Explicit results for KdV and Boussinesq hierarchies are provided.

Abstract

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.