Hodge-Frobenius equations and the Hodge-Backlund transformation

TL;DR

This paper explores Hodge-Frobenius equations and introduces Hodge-Backlund transformations, connecting integrable systems with classical transformations through variational principles and Hodge involution properties.

Contribution

It presents a novel approach to Hodge-like systems using integrability assumptions and develops Hodge-Backlund transformations linking different energy densities.

Findings

Hodge-Backlund transformations relate standard energy densities.

Invariant forms of classical Backlund transformations are derived.

Extensions to higher-degree forms are proposed.

Abstract

Linear and nonlinear Hodge-like systems for 1-forms are studied, with an assumption equivalent to complete integrability substituted for the requirement of closure under exterior differentiation. The systems are placed in a variational context and properties of critical points investigated. Certain standard choices of energy density are related by Backlund transformations which employ basic properties of the Hodge involution. These Hodge-Backlund transformations yield invariant forms of classical Backlund transformations that arise in diverse contexts. Some extensions to higher-degree forms are indicated.