Hierarchy of Hamilton equations on Banach Lie-Poisson spaces related to restricted Grassmannian

TL;DR

This paper develops a hierarchy of Hamilton equations on Banach Lie-Poisson spaces related to the restricted Grassmannian, including Ricatti-type equations, with explicit solutions for specific cases.

Contribution

It introduces a new hierarchy of Hamilton equations on Banach Lie-Poisson spaces associated with the restricted Grassmannian, expanding the understanding of their integrable structures.

Findings

Hierarchy of Hamilton equations constructed and analyzed.

Ricatti-type operator equations included in the hierarchy.

Explicit solutions provided for specific cases.

Abstract

Using the Magri method one defines an involutive family of Hamiltonians on Banach Lie-Poisson space iR+UL_res^1 (which contains the restricted Grassmannian as a symplectic leaf) and on its complexification C+L_res^1. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases one gives the explicit solutions.