Title: On Lie algebras responsible for zero-curvature representations of multicomponent (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.