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Isaac Newton Institute for Mathematical Sciences

A proof that every Darboux transformation of arbitrary order $d$ for the ``Darboux-Laplace'' equation is a composition of first-order Darboux transformations

Presenter: Ekaterina Shemyakova (SUNY, New Paltz, USA)


Darboux's famous book ``Lessons on the General Theory of Surfaces and the Geometric Applications of Infinitesimal Calculus'', 1887, is one of the foundations of differential geometry and the theory of integrable systems. This book contains many fundamental ideas. Unfortunately, some of them were neither stated in a rigorous form, nor proved.

In this talk we formulate and prove one of such ideas, which is fundamental for the theory of Darboux transformations. (As it is known, such transformations are an essential tool in the theory of integrable systems, see the latest developments in Grinevich, Novikov (2012), Grinevich, Mironov, Novikov (2011), Taimanov (2008).) Darboux expected that the transformations that now bear his name can be all generated by quasideterminants (the theory of which have been developed by Gelfand-Retakh in 90-ties) and the Laplace transformations. A few particular cases of this theorem have been proved or have been attempted to be proved by several different authors since 1995.

We prove it in the general case: every Darboux transformation of arbitrary order $d$ for the ``Darboux-Laplace'' equation \[ u_{xy} + a(x,y) u_x + b(x,y) u_y + c(x,y) u=0 \] is a composition of first-order Darboux transformations. Even the case of transformations of order two is not trivial.