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On the periodic Toda lattice with a self-consistent source. (English) Zbl 1329.37062
Summary: This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice with self-consistent source. The effective method of solution of the inverse spectral problem for the discrete Hill’s equation is presented.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] Toda, M., Waves in nonlinear lattice, Suppl Prog Theor Phys, 45, 74-200, (1970)
[2] Flashka, H., On the Toda lattice, II, Prog Theor Phys, 51, 703-716, (1974) · Zbl 0942.37505
[3] Manakov, S. V., Complete integrability and stochastization of discrete dynamical systems, Zh Eksper Teoret Fiz, 67, 543-555, (1974)
[4] Toda, M., Theory of nonlinear lattices, (1981), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0465.70014
[5] Dubrovin, B. A.; Matveev, V. B.; Novikov, S. P., Non-linear equations of Korteweg-de Vries type finite-zone linear operators and abelian varieties, Usp Mat Nauk, 31, 1(187), 55-136, (1976) · Zbl 0326.35011
[6] Date, E.; Tanaka, S., Analog of inverse scattering theory for discrete hill’s equation and exact solutions for the periodic Toda lattice, Prog Theor Phys, 55, 217-222, (1976)
[7] Krichever, I. M., Algebraic curves and non-linear difference equations, Usp Mat Nauk, 33, 4(202), 215-216, (1978) · Zbl 0382.39003
[8] Teshl, G., Jacobi operators and completely integrable lattices, Mathematical surveys and monographs, vol. 72, (2000), AMS
[9] Samoilenko, V. G.; Prikarpatskii, A. K., Periodic problem for a Toda chain, Ukrainian Math J, 34, 380-385, (1982)
[10] Melnikov, V. K., Exact solutions of the Korteweg-de Vries equation with a self-consistent source, Phys Lett A, 128, 488-492, (1988)
[11] Urazboev, G. U., Toda lattice with a special self-consistent source, Theor Math Phys, 154, 305-315, (2008) · Zbl 1155.37040
[12] Khasanov, A. B.; Yakhshimuratov, A. B., The Korteweg-de Vries equation with a self-consistent source in the class of periodic functions, Theor Mat Fiz, 164, 214-221, (2010) · Zbl 1301.35134
[13] Blackmore, D.; Prikarpatskii, A. K.; Samoilenko, V. G., Nonlinear dynamical systems of mathematical physics: spectral and symplectic integrability analysis, (2012), World Scientific Singapore
[14] Prikarpatskii, A. K.; Mykytiuk, I. V., Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects, (1998), Kluwer Academic Publisher · Zbl 0937.37055
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