A geometric framework to compare classical field theories and to transfer solutions between PDEs

TL;DR

This paper develops a geometric framework using jet spaces to compare and relate different classical field theories described by PDEs, enabling solution transfer and generating new solutions through generalized Bäcklund transformations.

Contribution

It introduces a novel geometric approach to compare PDE systems, derive Bäcklund transformations, and explicitly compute minimal conditions for shared structures in classical field theories.

Findings

Shared structures enable solution transfer between PDEs.

Generalized Bäcklund transformations generate new solutions.

Framework applied to electrodynamics and hydrodynamics.

Abstract

In this contribution, a mathematical framework is constructed to relate and compare non-linear partial differential equations (PDEs) in the category of smooth manifolds. In particular, it can be used to compare those aspects of field theories (e.g. of classical (Newtonian) mechanics, hydrodynamics, electrodynamics, relativity theory, classical Yang-Mills theory and so on) that are described by such equations. Employing a geometric (jet space) approach, a suitable notion of shared structure of two systems of PDEs is identified. It is proven that this shared structure can serve to transfer solutions from one theory to another and a generalization of so-called B\"acklund transformations is derived that can be used to generate non-trivial solutions of some non-linear PDEs. A procedure (based on formal integrability) is introduced with which one can explicitly compute the minimal consistency conditions that two systems of PDEs need to fulfill in order to share structure under a given correspondence. Furthermore, it is shown how symmetry groups can be used to identify useful correspondences and structure that is shared up to symmetries. Thereby, the role that B\"acklund transformations play in the theory of quotient equations is clarified. Explicit examples illustrate the general ideas throughout the text and in the last chapter, the framework is applied to systems related to electrodynamics and hydrodynamics.