On Lie algebras responsible for zero-curvature representations of multicomponent (1+1)-dimensional evolution PDEs

TL;DR

This paper develops methods to analyze the structure of Lie algebras associated with zero-curvature representations of multicomponent (1+1)-dimensional PDEs, enabling classification of ZCRs and aiding in the study of integrability and transformations.

Contribution

It introduces new techniques to explicitly compute and classify the Lie algebras governing ZCRs for complex PDE systems, extending understanding of their algebraic structure.

Findings

Computed the structure of F^p algebras for Landau-Lifshitz and nonlinear Schrödinger equations

Classified all ZCRs for the n-component Landau-Lifshitz system up to gauge equivalence

Provided methods applicable to other (1+1)-dimensional evolution PDEs

Abstract

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable $(1+1)$-dimensional PDEs. According to the preprint arXiv:1212.2199, for any given $(1+1)$-dimensional evolution PDE one can define a sequence of Lie algebras $F^p$, $p=0,1,2,3,\dots$, such that representations of these algebras classify all ZCRs of the PDE up to local gauge equivalence. ZCRs depending on derivatives of arbitrary finite order are allowed. Furthermore, these algebras provide necessary conditions for existence of Backlund transformations between two given PDEs. The algebras $F^p$ are defined in arXiv:1212.2199 in terms of generators and relations. In the present paper, we describe some methods to study the structure of the algebras $F^p$ for multicomponent $(1+1)$-dimensional evolution PDEs. Using these methods, we compute the explicit structure (up to non-essential nilpotent ideals) of the Lie algebras $F^p$ for the Landau-Lifshitz, nonlinear Schrodinger equations, and for the $n$-component Landau-Lifshitz system of Golubchik and Sokolov for any $n>3$. In particular, this means that for the $n$-component Landau-Lifshitz system we classify all ZCRs (depending on derivatives of arbitrary finite order), up to local gauge equivalence and up to killing nilpotent ideals in the corresponding Lie algebras. The presented methods to classify ZCRs can be applied also to other $(1+1)$-dimensional evolution PDEs. Furthermore, the obtained results can be used for proving non-existence of Backlund transformations between some PDEs, which will be described in forthcoming publications.