Speaker:  Sergei Igonin (Yaroslavl State University) 

Date and time:  19.06.2024, 17:00 (GMT +03:00)

Title: Discrete Miura-type transformations, gauge simplifications, and group actions associated with Lax pairs for differential-difference equations

Abstract:  In this talk I will discuss some relations between matrix differential-difference Lax pairs, gauge transformations, and discrete Miura-type transformations for differential-difference (lattice) equations.

I will present sufficient conditions for the possibility to simplify a matrix differential-difference Lax pair by local matrix gauge transformations.

Also, I will present a method to construct Miura-type transformations for differential-difference equations, using gauge transformations and invariants of Lie group actions on manifolds associated with Lax pairs of such equations.

The method is applicable to a wide class of Lax pairs.

The considered examples include the (modified) Volterra, Itoh-Narita-Bogoyavlensky, Toda lattice equations, a differential-difference discretization of the Sawada-Kotera equation, and Adler-Postnikov equations from [1]. Applying the method to these examples, one obtains new integrable nonlinear differential-difference equations connected with these equations by new Miura-type transformations.

Some steps of our method generalize (in the differential-difference setting) a result of V.G. Drinfeld and V.V. Sokolov [3] on Miura-type transformations for the partial differential KdV equation.

This talk is based on the preprint [4] and a joint paper with G. Berkeley [2].

[1] V.E. Adler, V.V. Postnikov. Differential-difference equations associated with the fractional Lax operators. J. Phys. A: Math. Theor. (2011) 44, 415203.

[2] G. Berkeley, S. Igonin. Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations. J. Phys. A: Math. Theor. (2016) 49, 275201.  https://arxiv.org/abs/1512.09123

[3] V.G. Drinfeld, V.V. Sokolov. On equations that are related to the Korteweg-de Vries equation. Soviet Math. Dokl. (1985) 32, 361-365.

[4] S. Igonin. Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations. (2024)  https://arxiv.org/abs/2403.12022

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com