1212.2199 (Sergey Igonin)

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

Sergey Igonin

For any (1+1)-dimensional evolution PDE, we define a sequence of Lie algebras $F^p$, $p=0,1,2,...$, 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 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. In the obtained algebras, one encounters solvable ideals, semisimple ideals, and infinite-dimensional Lie algebras of matrix-valued functions on algebraic curves. Applications to classification of KdV and Krichever-Novikov type equations with respect to Backlund transformations are also discussed.

View original: http://arxiv.org/abs/1212.2199