This article is cited in 1 paper

Nonlinear physics and mechanics

Inverse Spectral Problems for Second-Order Difference Operators and Their Application to the Study of Volterra Type Systems

A. S. Osipov

Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Nakhimovskii pr. 36-1, Moscow, 117218 Russia

Abstract: In this paper, some links between inverse problem methods for the second-order difference operators and nonlinear dynamical systems are studied. In particular, the systems of Volterra type are considered. It is shown that the classical inverse problem method for semi-infinite Jacobi matrices can be applied to obtain a hierarchy of Volterra lattices, and this approach is compared with the one based on Magri’s bi-Hamiltonian formalism. Then, using the inverse problem method for nonsymmetric difference operators (which amounts to reconstruction of the operator from the moments of its Weyl function), the hierarchies of Volterra and Toda lattices are studied. It is found that the equations of Volterra hierarchy can be transformed into their Toda counterparts, and this transformation can be easily described in terms of the above-mentioned moments.

Keywords: inverse spectral problems, difference operators, Jacobi matrices, Volterra lattices, Toda lattices.

MSC: 35Q58, 37J35, 47B39, 39A70

Received: 12.12.2019
Accepted: 23.07.2020

Language: English

DOI: 10.20537/nd200301

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