A new scheme of integrability for (bi)Hamiltonian PDE

TL;DR

This paper introduces a novel method for constructing integrable Hamiltonian hierarchies of Lax type equations by combining fractional powers and Hamiltonian reduction techniques, utilizing Adler type operators and generalized quasideterminants.

Contribution

It presents a new integrability scheme for (bi)Hamiltonian PDEs that unifies existing methods through the use of Adler type matrices and quasideterminants, extending to dispersionless and non-commutative cases.

Findings

Developed a new construction method for integrable Hamiltonian hierarchies.

Introduced the concept of dispersionless Adler type series.

Extended the framework to non-commutative Hamiltonian equations.

Abstract

We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.