Seminar talk, 16 September 2015

From Geometry of Differential Equations
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Speaker: Sergei Igonin

Title: On Darboux-Bäcklund transformations for PDEs and Miura type transformations for differential-difference equations

Recall that differential-difference equations are equations for functions of an integer variable and a real variable . Such equations involve integer shifts of and derivatives with respect to . The well-known Toda lattice equation is a typical example.

In the first part of my talk, I will review some known results on connections between differential-difference equations and Darboux-Bäcklund transformations of two-dimensional PDEs.

For a given PDE with a zero-curvature (Lax) representation, Darboux transformations allow one to obtain new solutions for the PDE from known solutions by solving some ODEs. (The resulting relation between known and new solutions is a Bäcklund transformation.)

Following the review paper

F. Khanizadeh, A.V. Mikhailov, Jing Ping Wang, Theor. Math. Phys. 177, 1606-1654 (2013), arXiv:1305.0588, Mi tmf8550

I will show how differential-difference equations arise from Darboux transformations of PDEs.

In particular, we will see how the Toda lattice arises from Darboux transformations of the nonlinear Schrödinger equation.

Motivated by this construction, one defines the notion of Darboux-Lax representations (DLRs) for differential-difference equations. DLRs play the role of zero-curvature representations in the differential-difference case. (This is also known.)

In the second part of my talk, I will present new results on Miura type transformations (MTTs) for differential-difference equations (DDEs). Namely, I will describe a method to construct MTTs for DDEs from DLRs of DDEs. The method uses some Lie group actions associated with DLRs and is applicable to parameter-dependent DLRs satisfying certain conditions. The main idea behind this method is closely related to the results of Drinfeld and Sokolov on MTTs for the partial differential KdV equation.

The second part of my talk is based on a joint work with George Berkeley from the University of Leeds.