Darboux-Laplace transformations and related things

Sergey Smirnov (University of Glasgow)

Thursday 21st August 16:00-17:00
Maths 311B

Abstract

Darboux transformations play an important role in the theory of integrable systems. We will consider a particular case of hyperbolic second-order operators, review some known facts about Darboux-Laplace transformations in the differential case and discuss an on-going work in the difference case. In the continuous case first-order DLTs are classified: the first class consists of classical Laplace transformations and the second class is given by Wronskian formulas. Iterations of transformations of the first type lead to two-dimensional Toda systems. If the Laplace series of the initial hyperbolic operator is finite, then the corresponding Toda lattice is Darboux integrable, and the family of all such operators is described by an elegant determinant formula (Darboux formula). In the (semi)-discrete case there is only a partial classification result for DLTs. The analogue of classical Laplace transformation leads to discrete 2D Toda lattice. We will discuss discrete versions of the Darboux determinant formula and its possible applications in Discrete Differential Geometry.

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