Algebras and algebraic curves associated with PDEs and Bäcklund transformations

Using the language of jet spaces, for any analytic PDE E we define, in a coordinatefree way, a family of associative algebras A(E). In the considered examples, which include the KdV, Krichever-Novikov, nonlinear Schr¨odinger, Landau-Lifshitz equations, the algebras A(E) are commutative and are isomorphic to the function field of an algebraic curve of genus 1 or 0. This provides an invariant meaning for algebraic curves related to some PDEs. Also, the algebras A(E) help to prove that some pairs of PDEs from the above list are not connected by B¨acklund transformations. To define A(E), we use fundamental Lie algebras F(E) of E introduced in [15]. Elements of A(E) are intertwining operators for the adjoint representations of Lie subalgebras of certain quotients of F(E).