B\"acklund Transformations an Zero-Curvature Representations of Systems of Partial Differential Equations

TL;DR

This paper explores the deep relationship between Bäcklund transformations and zero-curvature representations in PDEs, providing methods to construct one from the other and a systematic approach to find ZCRs.

Contribution

It establishes a connection between BTs and ZCRs via symmetry group representations and outlines a systematic search procedure for ZCRs of PDE systems.

Findings

BTs can be constructed from ZCRs using gauge transformations

A systematic method for finding ZCRs is proposed

The connection enhances understanding of integrability in PDEs

Abstract

It is shown that B\"acklund transformations (BTs) and zero-curvature representations (ZCRs) of systems of partial differential equations (PDEs) are closely related. The connection is established by nonlinear representations of the symmetry group underlying the ZCR which induce gauge transformations relating different BTs. This connection is used to construct BTs from ZCRs (and vice versa). Furthermore a procedure is outlined which allows a systematic search for ZCRs of a given system of PDEs.