Lie algebras associated with PDEs and Bäcklund transformations

Abstract

We introduce a new geometric invariant of PDEs: with any analytic system of PDEs we associate naturally a certain system of Lie algebras. These Lie algebras are responsible for Backlund transformations (a tool to construct exact solutions for nonlinear PDEs) and zero-curvature representations (including 2-dimensional Lax pairs) in the theory of integrable systems. Using infinite jet spaces, we regard PDEs as infinite-dimensional manifolds with involutive distributions and study their special morphisms called Krasilshchik-Vinogradov coverings, which generalize the classical concept of coverings from topology and provide a geometric framework for Backlund transformations, Lax pairs, and some other constructions in soliton theory. Recall that topological coverings of a manifold M can be described in terms of the fundamental group of M. We show that a similar description exists for finite-rank Krasilshchik-Vinogradov coverings of PDEs. However, the "fundamental group of a PDE" is not a group, but a certain system of Lie algebras, which we call fundamental algebras. We have computed these algebras for a number of well-known nonlinear PDEs. As a result, one obtains infinite-dimensional Lie algebras of Kac-Moody type and Lie algebras of matrix-valued functions on algebraic curves. Applications to construction and classification of Backlund transformations will be also presented.