Miura-like transformations between Bogoyavlensky lattices and inverse spectral problems for band operators

TL;DR

This paper explores transformations between Bogoyavlensky lattices and their connection to inverse spectral problems for band operators, revealing new insights into their integrability and first integrals.

Contribution

It introduces Miura-like transformations for Bogoyavlensky lattices and links them to inverse spectral methods using moments of Weyl matrices.

Findings

Transformations described via Weyl matrix moments

Lattices shown to be integrable through inverse spectral methods

Finite lattices' first integrals characterized

Abstract

We consider semi-infinite and finite Bogoyavlensky lattices \begin{eqnarray*} \overset\cdot a_i&=&a_i\left(\prod_{j=1}^{p}a_{i+j}-\prod_{j=1}^{p}a_{i-j}\right),\\ \overset\cdot b_i&=&b_i\left(\sum_{j=1}^{p} b_{i+j}-\sum_{j=1}^{p}b_{i-j}\right), \end{eqnarray*} for some $p\ge 1,$ and Miura-like transformations between these systems, defined for $p\ge 2$. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i. e. operators generated by (possibly infinite) 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 the finite Bogoyavlensky lattices and in particular their first integrals.