New global solutions of the Jacobi partial differential equations

TL;DR

This paper introduces a new family of solutions to the Jacobi partial differential equations in finite-dimensional Poisson systems, highlighting their mathematical properties and potential for generating infinite solution families.

Contribution

It presents a novel class of distinguished solutions (D-solutions) with unique properties, expanding the understanding of Poisson structures and their solution space.

Findings

D-solutions are associated with invariants of the solutions themselves.

A constructive family of D-solutions is characterized and analyzed.

Applications include the global construction of Darboux canonical form.

Abstract

A new family of solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is investigated. This family is mathematically remarkable, as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves. This kind of Poisson structure (termed distinguished solutions or D-solutions) is defined for every nontrivial combination of values of the dimension and the rank, and is also determined in terms of functions of arbitrary nonlinearity, properties usually not present simultaneously in the already known solution families. In addition, D-solutions display several properties allowing the generation of an infinity of D-solutions from a given one, which is an uncommon feature in the framework of the Jacobi equations. Furthermore, a special family of D-solutions complying with the previous requirements is constructively characterized and analyzed. Examples are discussed focusing on physical implications and including an application for the global construction of the Darboux canonical form.