Abstract
A new category of topological jet manifolds is proposed for the purpose of investigating exact finite-dimensional approximations of nonlinear dynamical systems on infinite-dimensional functional manifolds. Differential geometry structures on these manifolds and their applications to the theory of integrability in quadratures of nonlinear dynamical Lax-type systems are studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. P. Novikov (ed.), Theory of Solitons [in Russian], Nauka, Moscow (1980), 342 pp.
L. A. Takhtadzhyan and L. D. Fadeev, Hamiltonian Approach to Theory of Solitons [in Russian], Nauka, Moscow (1986), 527 pp.
Yu. A. Mitropol'skii [Mytropol's'kyy], N. N. Bogolyubov, Jr., A. K. Prikarpatskii [Prykarpats'kii], and V. G. Samoilenko, Integrable Dynamical Systems [in Russian], Nauk. Dumka, Kiev (1987), 296 pp.
V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1979), 431 pp.
V. G. Samoilenko, Jet Analysis on Smooth Infinite-Dimensional Manifolds and Its Application to Investigations of the Integrability of Nonlinear Dynamical Systems [in Russian], Preprint No. 88.51, Institute of Mathematics of the Academy of Sciences of Ukrainian SSR, Kiev (1988), 23 pp.
V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Manifolds [in Russian], Nauka, Moscow (1982), 340 pp.
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1984), 710 pp.
M. M. Pritula, A. K. Prykarpats'kii, and I. V. Mykytyuk, Elements of the Theory of Differential Geometry Structures and Dynamical Systems [in Ukrainian], UMKVO, Kiev (1988), 87 pp.
P. Oevel, Applications of Lie Groups to Differential Equations [Russian translation], Mir, Moscow (1990), 536 pp.
A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Introduction to the Geometry of Nonlinear Differential Equations [in Russian], Nauka, Moscow (1986), 335 pp.
M. Gromov, Partial Differential Relations [Russian translation], Mir, Moscow (1990), 536 pp.
Yu. O. Mytropol's'kyy, A. K. Prykarpats'kyy, and B. M. Fil', “Certain aspects of the gradient-holonomy algorithm to the theory of integrability of nonlinear dynamical systems and problems of computer algebra,” Ukr. Mat. Zh.,43, No. 1, 78–91 (1991).
A. K. Prikarpatskii and I. V. Mikiyuk [Mykytyuk], Algebraic Aspects of Integrability of Nonlinear Dynamical Systems on Manifolds [in Russian], Nauk. Dumka, Kiev (1991), 267 pp.
A. Yu. Khrennikov, “Functional superanalysis,” Usp. Mat. Nauk,43, No. 2, 87–114 (1988).
D. Kastler, “The Koszul formula for graded Lie-Cartan pairs (super BBS operator),” J. Geom. and Phys.,4, No. 4, 523–534 (1987).
A. Jadezyk and D. Kastler, “Graded Lie-Cartan pairs. II. The fermionic differential calculus,” Ann. Phys.,179, No. 2, 169–200 (1987).
R. Trostel, “Color analysis, theory ofΓ-graded integrable evolution equations, and super-Nijenhuis operators,” J. Math. Phys.,26, No. 12, 3160–3171 (1985).
Yu. I. Manin and A. O. Radul, “A supersymmetric extension of the Kadomtsev-Petviashwili hierarchy,” Communs. Math. Phys.,98, No. 1, 65–77 (1985).
K. Gawedzki, “Supersymmetric extension of the Korteweg-de Vries equation,” Ann. Inst. H. Poincaré. A,27, No. 4, 335–366 (1977).
P. Mathieu, “Supersymmetric extension of the Korteweg-de Vries equation,” J. Math. Phys.,29, No. 11, 2499–2506 (1988).
P. P. Kulish, “Analog of Korteweg-de Vries equation for a superconformal algebra,” Differents. Geom., Gruppy Li i Mekhanika,101, 64–68 (1987).
W. Oevel, “R-structures, Yang-Baxter equations, and related involution theorems,” J. Math. Phys.,30, No. 5, 1140–1149 (1989).
B. N. Fil', Supergeneralization of Fully Integrable Dynamical Systems [in Russian], Preprint No. 89.28, Institute of Mathematics of the Academy of Sciences of Ukrainian SSR, Kiev (1989), 29 pp.
R. A. Chowdhury. and S. Roy, “On the Bäcklund transformation and Hamiltonian properties of superevaluation equations,” J. Math. Phys.,27, No. 10, 2464–2468 (1986).
L. Yi-Shen and L.-N. Zhang, “Super AKNS-scheme and its finite conserved currents,” Nuovo cim. A,93, No. 2, 175–183 (1986).
M. Gurses and O. Oguz, “A super AKNS-scheme,” Phys. Lett. A,108, No. 9, 437–440 (1985).
B. A. Kupershmidt, “Integrable systems,” Proc. Nat. Acad. Sci. USA, No. 81, 6562–6563 (1984).
Author information
Authors and Affiliations
Additional information
Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1242–1256, September, 1992.
Rights and permissions
About this article
Cite this article
Prykarpats'kyy, A.K., Fil', B.M. Category of topological jet manifolds and certain applications in the theory of nonlinear infinite-dimensional dynamical systems. Ukr Math J 44, 1136–1148 (1992). https://doi.org/10.1007/BF01058376
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01058376