Higher jet prolongation Lie algebras and Backlund transformations for (1+1)-dimensional PDEs

TL;DR

This paper introduces a hierarchy of Lie algebras associated with (1+1)-dimensional PDEs, which classify all Lax pairs and zero-curvature representations, aiding in understanding Backlund transformations and PDE integrability.

Contribution

It defines a sequence of Lie algebras $F^p$ for arbitrary jet order, generalizing Wahlquist-Estabrook algebras, and applies them to classify PDEs via Backlund transformations.

Findings

Computed $F^p(E,a)$ for KdV-type equations.

Found infinite-dimensional and semisimple Lie algebras within $F^p(E,a)$.

Established a necessary condition for PDEs to be connected by Backlund transformations.

Abstract

For any (1+1)-dimensional (multicomponent) evolution PDE, we define a sequence of Lie algebras $F^p$, $p=0,1,2,3,...$, which are responsible for all Lax pairs and zero-curvature representations (ZCRs) of this PDE. In our construction, jets of arbitrary order are allowed. In the case of lower order jets, the algebras $F^p$ generalize Wahlquist-Estabrook prolongation algebras. To achieve this, we find a normal form for (nonlinear) ZCRs with respect to the action of the group of gauge transformations. One shows that any ZCR is locally gauge equivalent to the ZCR arising from a vector field representation of the algebra $F^p$, where $p$ is the order of jets involved in the $x$-part of the ZCR. More precisely, we define a Lie algebra $F^p$ for each nonnegative integer $p$ and each point $a$ of the infinite prolongation $E$ of the evolution PDE. So the full notation for the algebra is $F^p(E,a)$. Using these algebras, one obtains a necessary condition for two given evolution PDEs to be connected by a Backlund transformation. In this paper, the algebras $F^p(E,a)$ are computed for some PDEs of KdV type. In a different paper with G. Manno, we compute $F^p(E,a)$ for multicomponent Landau-Lifshitz systems of Golubchik and Sokolov. Among the obtained Lie algebras, one encounters infinite-dimensional algebras of certain matrix-valued functions on some algebraic curves. Besides, some solvable ideals and semisimple Lie algebras appear in the description of $F^p(E,a)$. Applications to classification of KdV and Krichever-Novikov type equations with respect to Backlund transformations are also briefly discussed.