Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications

TL;DR

This paper develops Darboux matrices and explicit solutions for generalized Hamiltonian systems with rational spectral dependence, enabling new insights and solutions for multi-variable dynamical systems.

Contribution

It introduces a novel approach using Bäcklund-Darboux transformations and matrix eigenvalue roots to explicitly solve generalized Hamiltonian systems with spectral parameter dependence.

Findings

Constructed Darboux matrices for Hamiltonian systems

Derived explicit solutions for multi-variable dynamical systems

Included new auxiliary results on matrix roots

Abstract

We construct so called Darboux matrices and fundamental solutions in the important case of the generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in this way. Interesting results for dynamical systems depending on several variables and their explicit solutions follow. For these purposes we use our version of B\"acklund-Darboux transformation and square roots of the corresponding generalised matrix eigenvalues. Some new auxiliary results on the roots of matrices are included as well. An appendix is added to make the paper self-sufficient.