Lie algebras responsible for zero-curvature representations of scalar evolution equations

TL;DR

This paper introduces a family of Lie algebras associated with scalar evolution equations that classify all zero-curvature representations, providing a framework to analyze integrability and transformations of PDEs.

Contribution

It defines the Lie algebras F(E) responsible for all ZCRs of scalar PDEs, generalizing previous algebraic structures and enabling classification and analysis of integrability.

Findings

Defined the Lie algebras F(E) for scalar evolution equations.

Established a normal form for ZCRs under gauge transformations.

Connected the structure of F(E) to integrability and Bäcklund transformations.

Abstract

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. For any (1+1)-dimensional scalar evolution equation $E$, we define a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations. As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a B\"acklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for B\"acklund transformations are presented in other publications as well. In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs. In this paper we describe general properties of $F(E)$ and present generators and relations for these algebras. In other publications we study the structure of $F(E)$ for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.