Hamiltonian systems and Sturm-Liouville equations: Darboux transformation and applications

TL;DR

This paper develops a generalized Darboux transformation for Hamiltonian and Sturm-Liouville systems, enabling explicit solution construction and transformation of associated functions, with novel results for general Hamiltonian and Shin-Zettl systems.

Contribution

It introduces the GBDT version of Darboux transformation for broad classes of systems, a first in the field, and applies it to explicit solution construction and function transformation.

Findings

First Darboux transformation results for general Hamiltonian systems

Explicit solutions for dynamical symplectic and Sturm-Liouville systems

Transformation of Weyl-Titchmarsh functions using GBDT

Abstract

We introduce GBDT version of Darboux transformation for symplectic and Hamiltonian systems as well as for Shin-Zettl systems and Sturm-Liouville equations. These are the first results on Darboux transformation for general-type Hamiltonian and for Shin-Zettl systems. The obtained results are applied to the corresponding transformations of the Weyl-Titchmarsh functions and to the construction of explicit solutions of dynamical symplectic systems, of two-way diffusion equations and of indefinite Sturm-Liouville equations. The energy of the explicit solutions of dynamical systems is expressed (in a quite simple form) in terms of the parameter matrices of GBDT.