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

Voir la notice de l'article provenant de la source Math-Net.Ru

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 
@article{ND_2020_16_3_a0,
     author = {A. S. Osipov},
     title = {Inverse {Spectral} {Problems} for {Second-Order}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {397--419},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_3_a0/}
}
TY  - JOUR
AU  - A. S. Osipov
TI  - Inverse Spectral Problems for Second-Order
JO  - Russian journal of nonlinear dynamics
PY  - 2020
SP  - 397
EP  - 419
VL  - 16
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2020_16_3_a0/
LA  - en
ID  - ND_2020_16_3_a0
ER  - 
%0 Journal Article
%A A. S. Osipov
%T Inverse Spectral Problems for Second-Order
%J Russian journal of nonlinear dynamics
%D 2020
%P 397-419
%V 16
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2020_16_3_a0/
%G en
%F ND_2020_16_3_a0
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/

[1] Kac, M. and van Moerbeke, P., “On an Explicitly Soluble System of Nonlinear Differential Equations Related to a Certain Toda Lattice”, Adv. Math., 16:2 (1975), 160–169 | DOI | MR | Zbl

[2] Moser, J., “Three Integrable Hamiltonian Systems Connected with Isospectral Deformations”, Adv. Math., 16:2 (1975), 197–220 | DOI | MR | Zbl

[3] Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Oliver Boyd, Edinburgh, 1965, 253 pp. | MR | Zbl

[4] Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space, Dower, New York, 1993 | MR

[5] Babelon, O., “Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket”, Comm. Math. Phys., 266:3 (2006), 819–862 | DOI | MR | Zbl

[6] Berezanskii, Ju. M., Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr., 17, AMS, Providence, R.I., 1968, ix+809 pp. | DOI | MR

[7] Berezanski, Yu. M., “The Integration of Semi-Infinite Toda Chain by Means of Inverse Spectral Problem”, Rep. Math. Phys., 24:1 (1986), 21–47 | DOI | MR

[8] Barrios Rolanía, D., Branquinho, A., and Foulquié Moreno, A., “Dynamics and Interpretation of Some Integrable Systems via Multiple Orthogonal Polynomials”, J. Math. Anal. Appl., 361:2 (2010), 358–370 | DOI | MR | Zbl

[9] Bogoyavlensky, O. I., Breaking Solitons. Nonlinear Integrable Equations, Nauka, Moscow, 1991, 320 pp. (Russian) | MR

[10] Damianou, P. A., “The Volterra Model and Its Relation to the Toda Lattice”, Phys. Lett. A, 155:2–3 (1991), 126–132 | DOI | MR

[11] Damianou, P. A. and Loja Fernandes, R., “From the Toda Lattice to the Volterra Lattice and Back”, Rep. Math. Phys., 50:3 (2002), 361–378 | DOI | MR | Zbl

[12] Osipov, A., “On Some Issues Related to the Moment Problem for the Band Matrices with Operator Elements”, J. Math. Anal. Appl., 275:2 (2002), 657–675 | DOI | MR | Zbl

[13] Osipov, A. S., “On a Determinant Criterion for the Weak Perfectness of Systems of Functions Holomorphic at Infinity and Their Connection with Some Classes of Higher-Order Difference Operators”, Fundam. Prikl. Mat., 1:3 (1995), 711–727 (Russian) | MR | Zbl

[14] Mat. Zametki, 56:6 (1994), 141–144 (Russian) | DOI | MR | Zbl

[15] Magri, F., “A Simple Model of the Integrable Hamiltonian Equation”, J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[16] Morosi, C. and Pizzocchero, L., “On the Continuous Limit of Integrable Lattices: 1. The Kac – Moerbeke System and KdV Theory”, Comm. Math. Phys., 180:2 (1996), 505–528 | DOI | MR | Zbl

[17] Yurko, V. A., “On Higher-Order Difference Operators”, J. Differ. Equations Appl., 1:4 (1995), 347–352 | DOI | MR | Zbl

[18] Gantmacher, F. R., The Theory of Matrices, v. 2, Chelsea, New York, 2000, 288 pp. | MR

[19] Gekhtman, M., “Hamiltonian Structure of Non-Abelian Toda Lattice”, Lett. Math. Phys., 46:3 (1998), 189–205 | DOI | MR | Zbl

[20] Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 150 (1986), 17–25, 218 (Russian) | DOI | MR | Zbl | Zbl