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

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For any (1+1)-dimensional (multicomponent) evolution PDE, we define a sequence of Lie algebras Fᵖ, 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 Fᵖ 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 Fᵖ, where p is the order of jets involved in the x-part of the ZCR. More precisely, we define a Lie algebra Fᵖ 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 Fᵖ(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 Fᵖ(E,a) are computed for some PDEs of KdV type. In a different paper with G. Manno, we compute Fᵖ(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 Fᵖ(E,a). Applications to classification of KdV and Krichever-Novikov type equations with respect to Backlund transformations are also briefly discussed.