Abstract
Isospectral problems for operator-valued Sturm-Liouville and Dirac differential expressions are considered. Within the framework of the gradient method, one establishes the complete integrability of the Lax associated nonlinear Hamiltonian systems with a bilocal implectic pair of Noetherian operators on a manifold of integral operators.
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Literature cited
S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Consultants Bureau, New York (1984).
Yu. A. Mitropol'skii, N. N. Bogolyubov, Jr., A. K. Prikarpatskii, and V. G. Samoilenko, Integrable Dynamic Systems: Spectral and Differential-Geometric Aspects [in Russian], Naukova Dumka, Kiev (1987).
A. S. Fokas and P. M. Santini, “Bi-Hamiltonian formulation of the Kadomtsev-Petviashvili and Benjamin-Ono equations,” J. Math. Phys.,29, No. 3, 604–617 (1988).
P. M. Santini and A. S. Fokas, “Recursion operators and bi-Hamiltonian structures in multidimensions. I,” Commm. Math. Phys.,115, No. 3, 375–419 (1988).
A. S. Fokas and P. M. Santini, “The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schrödinger operator,” Stud. Appl. Math.,75, No. 2, 179–186 (1986).
N. N. Bogolyubov, Jr. and A. K. Prikarpatskii, “Quantum Lie algebra of currents as the universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems in theoretical and mathematical physics,” Teor. Mat. Fiz.,75, No. 1, 3–17 (1988).
V. G. Samoilenko and B. N. Fil', “Algebras of symmetries of completely integrable dynamical systems,” Ukr. Mat. Zh.,40, No. 2, 192–198 (1988).
A. P. Veselov and S. P. Novikov, “The integrability of the two-dimensional KdV,” Dokl. Akad. Nauk SSSR,197, No. 5, 705–708 (1984).
V. I. Gorbachuk and M. L. Gorbachuk, Spectral Theory of Operator Boundary Problems [in Russian], Naukova Dumka, Kiev (1985).
M. Boiti, H. J.-P. Léon, and F. Pempinelli, “Canonical and noncanonical recursion operators in multidimensions,” Stud. Appl. Math.,78, No. 1, 1–19 (1988).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 794–800, June, 1990.
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Bogolyubov, N.N., Prikarpatskii, A.K. A bilocal periodic problem for the Sturm-Liouville and Dirac operators and some applications to the theory of nonlinear dynamical systems. I. Ukr Math J 42, 702–707 (1990). https://doi.org/10.1007/BF01058917
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DOI: https://doi.org/10.1007/BF01058917