Hamiltonian systems with several space variables: dressing, explicit solutions and energy relations

TL;DR

This paper develops Darboux transformations for multi-variable Hamiltonian systems, providing explicit solutions and energy relations, with applications to port-Hamiltonian systems.

Contribution

It introduces a method to construct solutions and energy relations for multi-variable Hamiltonian systems, extending the theory of Darboux transformations.

Findings

Explicit solutions for specific Hamiltonian systems are provided.

Energy relations for these systems are derived.

The method is illustrated with several examples.

Abstract

We construct so-called Darboux transformations and solutions of the dynamical Hamiltonian systems with several space variables $\frac{\partial \psi}{\partial t}=\sum_{k=1}^r H_k(t)\frac{\partial \psi}{\partial \zeta_k}\,$ $( H_k(t)= H_k(t)^*)$. In particular, such systems are analogs of the port-Hamiltonian systems in the important and insufficiently studied case of several space variables. The corresponding energy relations are written down. The method is illustrated by several examples, where explicit solutions are given.