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

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.

💡 Research Summary

The paper develops a comprehensive algebraic framework for (1+1)-dimensional multicomponent evolution partial differential equations (PDEs) by introducing a hierarchy of Lie algebras denoted (F^{p}(E,a)), where (p=0,1,2,\dots). Here (E) is the infinite jet prolongation of the given PDE and (a) is a point on this infinite‑dimensional manifold. For each non‑negative integer (p), the algebra (F^{p}(E,a)) encodes all zero‑curvature representations (ZCRs) whose (x)-part involves jets of order at most (p). When (p) is low, these algebras coincide with the classical Wahlquist‑Estabrook prolongation algebras; the novelty lies in allowing arbitrarily high jet orders, thereby capturing a far richer class of Lax pairs.

A central technical achievement is the establishment of a normal‑form theorem for ZCRs under the action of the gauge group. The authors prove that any (possibly nonlinear) ZCR is locally gauge‑equivalent to a ZCR that arises from a vector‑field representation (\rho: F^{p}\to\mathfrak{X}(U)). In this normal form the connection 1‑form ((A,dx+B,dt)) satisfies the flatness condition (\partial_{t}A-\partial_{x}B+