AKS hierarchy and bi-Hamiltonian geometry of Gelfand–Zakharevich type

Abstract

A bi-Hamiltonian system is a system of differential equations which can be written in Hamiltonian form in two distinct ways. The applications of Gelfand–Zakharevich bi-Hamiltonian structure, which is an extension of a Poisson–Nijenhuis structure on phase space, has been extensively explored by Falqui, Magri, and Pedroni in the context of separation of variables. It is well known that the integrable Hamiltonian systems defined by the Adler–Kostant–Symes (AKS) scheme contains bi-Hamiltonian structure. In this paper we unveil the connection between Adler–Kostant–Symes formalism applied to loop algebra and the Gelfand–Zakharevich bi-Hamiltonian structure by superposition the results of Fordy and Kulish in the AKS scheme. We also study the commuting flows of the AKS hierachy and its connection to the Zakharov–Shabat hierarchy.