An Inverse Spectral Problem for a Special Class of Band Matrices and Bogoyavlensky Lattice

Osipov, Andrey

doi:10.1134/s1061920823010065

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Concluding Remarks and Open Issues1

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2024

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“…The operator L, which appears in the Lax representation for (68), differs sufficiently from the above operators L1, L2 and M , and the inverse spectral problems for such operators are less studied. Some recent results in this area are contained in [27]. The Miura transformation between (68) and (2)…”

Section: Concluding Remarks and Open Issuesmentioning

confidence: 99%

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

Osipov

2024

Armen.J.Math.

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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.

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Section: Concluding Remarks and Open Issuesmentioning

confidence: 99%