Andrey Osipov
Nakhimovskii pr. 36-1, Moscow, 117218 Russia
Scientific Research Institute for System Analysis of the Russian Academy of Sciences
Publications:
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Osipov A. S.
Inverse Spectral Problems for Second-Order Difference Operators and Their Application to the Study of Volterra Type Systems
2020, Vol. 16, no. 3, pp. 397-419
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 itsWeyl 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.
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