Geodesic
Inverse Spectral Problems for Second-Order
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 397-419

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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, Volterra lattices, Toda lattices.
Mots-clés : Jacobi matrices 
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     author = {A. S. Osipov},
     title = {Inverse {Spectral} {Problems} for {Second-Order}},
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     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_3_a0/}
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A. S. Osipov. Inverse Spectral Problems for Second-Order. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 397-419. http://geodesic.mathdoc.fr/item/ND_2020_16_3_a0/