Integrable Systems

10KP–KdV Hierarchy and Pseudo-differential Operators

In this chapter, we will study some generalities on the algebra of infinite-order differential operators. The algebras of Virasoro, Heisenberg and nonlinear evolution equations such as the Korteweg–de Vries (KdV), Boussinesq and Kadomtsev–Petviashvili (KP) equations play a crucial role in this study. We will make a careful study of a connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato–Date–Jimbo–Miwa–Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations, the Gelfand–Dickey flows, the Heisenberg and Virasoro algebras are also given.

10.1. Pseudo-differential operators and symplectic structures

Let L be a pseudo-differential operator with holomorphic coefficients. The set of these operators form a Lie algebra that we note 𝒜. The algebra 𝒜 decomposes in two subalgebras 𝒜₊ and 𝒜₋: 𝒜 = 𝒜₊ ⊕ 𝒜₋, where 𝒜₊ is the algebra of differential operators of the form , finite sum, , and 𝒜₋ is the algebra of strictly pseudo-differential operators of the form

The algebra 𝒜 can be seen as an associative algebra for the product of two pseudo-differential ...