Armenian Journal of Mathematics, 16(2)
2024; 1–28
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On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators

Andrey Osipov*

Received: 2024-12-03 · Published: 2024-02-13

© 2024 The Author(s) and the original publisher.

CC BY 4.0 This work is licensed under a Creative Commons Attribution 4.0 International License.
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Original title
On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators
Author
Andrey Osipov
Source journal
Armenian Journal of Mathematics, 16(2)
Published
2024-02-13
Licence
Creative Commons Attribution 4.0 International
Original
https://doi.org/10.52737/18291163-2024.16.2-1-28

Abstract

We consider semi-infinite and finite Bogoyavlensky lattices ⋅ai=ai(p∏j=1ai+j−p∏j=1ai−j),a⋅i=ai(∏j=1pai+j−∏j=1pai−j), ⋅bi=bi(p∑j=1bi+j−p∑j=1bi−j),b⋅i=bi(∑j=1pbi+j−∑j=1pbi−j), for some p≥1p≥1, and Miura-like transformations between these systems, defined for p≥2p≥2. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i.e., operators generated by band matrices. The key role in this method is played by the moments of the Weyl matrix of the corresponding band operator and their evolution in time. We find a description of the above-mentioned transformations in terms of these moments and apply this result to study finite Bogoyavlensky lattices and, in particular, their first integrals.
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