Igonin S. Lie algebras and algebraic curves responsible for Backlund transformations of PDEs, talk at ESF Exploratory Workshop EW10-078 (Vietri sul Mare, Italy), 13-16 June 2011 (abstract)
Speaker: Sergei Igonin
Title: Lie algebras and algebraic curves responsible for Bäcklund transformations of PDEs
We introduce a new geometric invariant of PDEs: with any analytic system of PDEs we associate naturally a certain system of Lie algebras.
Using infinite jet spaces, one can regard PDEs as geometric objects (manifolds with distributions) and obtains a category of PDEs. We study a special kind of morphisms in this category: the Krasilshchik-Vinogradov coverings, which generalize the classical concept of coverings from topology. They provide a geometric framework for Bäcklund transformations, which are a well-known tool to construct exact solutions for nonlinear PDEs.
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. In particular, these algebras are responsible for Bäcklund transformations and zero-curvature representations (including 2-dimensional Lax pairs) in the theory of integrable systems.
We have computed these algebras for several 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.
Using fundamental algebras, we obtain also an invariant meaning for algebraic curves related to some PDEs. Applications to construction and classification of Bäcklund transformations will be presented as well.
Slides: Igonin S. Lie algebras and algebraic curves responsible for Backlund transformations of PDEs (presentation at ESF Exploratory Workshop EW10-078, Vietri sul Mare, Italy, 13-16 June 2011).pdf
Sergei Igonin, Algebras and algebraic curves associated with PDEs and Bäcklund transformations, Max-Planck-Institut für Mathematik Preprint 120 (2010)