The Maxwell equations as a B\"acklund transformation

TL;DR

This paper explores how Maxwell's equations can be interpreted as a Bäcklund transformation, revealing a novel perspective on their integrability and connection to wave equations in electromagnetism.

Contribution

It demonstrates that Maxwell's equations serve as a Bäcklund transformation relating electric and magnetic wave equations, highlighting their linear system's integrability properties.

Findings

Maxwell equations can be viewed as a Bäcklund transformation.

The BT property holds for both vacuum and conducting media.

This perspective links linear PDEs in physics to nonlinear integrability techniques.

Abstract

Backlund transformations (BTs) are a useful tool for integrating nonlinear partial differential equations (PDEs). However, the significance of BTs in linear problems should not be ignored. In fact, an important linear system of PDEs in Physics, namely, the Maxwell equations of Electromagnetism, may be viewed as a BT relating the wave equations for the electric and the magnetic field, these equations representing integrability conditions for solution of the Maxwell system. We examine the BT property of this system in detail, both for the vacuum case and for the case of a linear conducting medium.