Integrable Hamiltonian systems related to the Hilbert--Schmidt ideal

TL;DR

This paper constructs hierarchies of integrable Hamiltonian systems on Banach Lie-Poisson spaces of Hilbert-Schmidt operators using coinduction and Magri methods, analyzing their properties and solutions in low dimensions.

Contribution

It introduces new integrable Hamiltonian hierarchies related to the Hilbert-Schmidt ideal and explores their algebraic, analytic properties, and explicit solutions.

Findings

Constructed integrable hierarchies on Banach Lie-Poisson spaces

Analyzed algebraic and analytic properties of these systems

Provided explicit solutions in dimensions N=2,3,4

Abstract

By application of the coinduction method as well as Magri method to the ideal of real Hilbert-Schmidt operators we construct the hierarchies of integrable Hamiltonian systems on the Banach Lie-Poisson spaces which consist of these type of operators. We also discuss their algebraic and analytic properties as well as solve them in dimensions N=2,3,4.