On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators

Abstract

We consider semi-infinite and finite Bogoyavlensky lattices
$$
\overset\cdot a_i =a_i\left(\prod_{j=1}^{p}a_{i+j}-\prod_{j=1}^{p}a_{i-j}\right),
$$
$$
\overset\cdot b_i = b_i\left(\sum_{j=1}^{p}
b_{i+j}-\sum_{j=1}^{p}b_{i-j}\right),
$$
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 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.