Ekaterina Shemyakova (State University of New York at New Paltz)
The aim of the talk is to give an introduction to the theory of Darboux-Laplace transformations (first part) as well as to present a proof (second part) of a long term conjecture hinted by Darboux at the end of 19th century.
Darboux transformations are the major tool in modern theory of integrable systems (soliton theory). Invented by Darboux in the context of classical differential geometry, these transformations have been rediscovered in 1970s. Many different versions have been since introduced. The general theory has not been settled yet.
In the second part of the talk we prove that for a hyperbolic partial differential operator or order $2$ on the plane every Darboux transformation of arbitrary order $d$ is a product of elementary Darboux transformations of order $1$.
The analogous statement for $1$-dimensional Schrödinger operator was proved earlier in four steps (Shabat, Veselov and Bagrov, Samsonov). In this case the factorization is not unique, and different factorizations imply discrete symmetries related to the Yang-Baxter maps (Adler and Veselov).