B\"acklund Transformations: Some Old and New Perspectives

TL;DR

This paper reviews Bäcklund transformations, highlighting their traditional role in solving nonlinear PDEs and exploring new applications like constructing recursion operators and solving linear PDE systems, exemplified by Maxwell's equations.

Contribution

It provides a comprehensive review of BTs, including recent extensions to linear PDEs and their connection to recursion operators, with detailed examples such as Maxwell's equations.

Findings

Maxwell equations form a Bäcklund transformation connecting electric and magnetic wave equations

Plane-wave solutions for Maxwell's equations are explicitly constructed

BTs are linked to recursion operators for symmetries of PDEs

Abstract

B\"acklund transformations (BTs) are traditionally regarded as a tool for integrating nonlinear partial differential equations (PDEs). Their use has been recently extended, however, to problems such as the construction of recursion operators for symmetries of PDEs, as well as the solution of linear systems of PDEs. In this article, the concept and some applications of BTs are reviewed. As an example of an integrable linear system of PDEs, the Maxwell equations of electromagnetism are shown to constitute a BT connecting the wave equations for the electric and the magnetic field; plane-wave solutions of the Maxwell system are constructed in detail. The connection between BTs and recursion operators is also discussed.