The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension

• Autorzy: Anatoliy Samoilenko , Yarema Prykarpatsky , Anatoliy Prykarpatsky
• Języki publikacji: EN

Abstrakty

EN

The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed.

Słowa kluczowe

EN

Delsarte transmutation operators; parametric functional spaces; Darboux transformations; inverse spectral transform problem; soliton equations; generalized de Rham-Hodge differential complex; Zakharov-Shabat equations; Laplace and Dirac type operators; 34A30; 34B05; 34B15

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 3
• Strony: 529-557

Daty

wydano

2005-09-01

online

2005-09-01

Twórcy

autor

Anatoliy Samoilenko

autor

Yarema Prykarpatsky

autor

Anatoliy Prykarpatsky

• The AGH University of Science and Technology

Bibliografia

• [1] J. Delsarte: “Sur certaines transformations fonctionelles relative aux equations lineaires aux derives partielles du second ordre”, C.R. Acad. Sci. Paris, Vol. 206, (1938), pp. 178–182.
• [2] J. Delsarte and J. Lions: “Transmutations d'operateurs differentielles dans le domain complex”, Comment. Math. Helv., Vol. 52, (1957), pp. 113–128.
• [3] I.V. Skrypnik: “Periods of A-closed forms”, Proceedings of the USSR Academy of Sciences, Vol. 160(4), (1965), pp. 772–773 (in Russian).
• [4] I.V. Skrypnik: “A harmonic fields with peculiarities”, Ukr. Math. Journal., Vol. 17(4), (1965), pp. 130–133 (in Russian).
• [5] I.V. Skrypnik: “The generalized De Rham theorem”, Proceed. of UkrSSR Acad. of Sci., Vol. 1, (1965), pp. 18–19 (in Ukrainian).
• [6] I.V. Skrypnik: “A harmonic forms on a compact Riemannian space”, Proceed. of UkrSSR Acad. of Sci., Vol. 2, p. 174–175 (in Ukrainian).
• [7] Y.B. Lopatynski: “On harmonic fields on Riemannian manifolds”, Ukr. Math. Journal, Vol. 2, (1950), pp. 56–60 (in Russian).
• [8] S.S. Chern: Complex manifolds, Chicago University Publ., USA, 1956.
• [9] L.D. Faddeev: “Quantum inverse scattering problem. II”, In: Modern problems of mathematics, Vol. 3, M: VINITI Publ., 1974, pp. 93–180 (in Russian).
• [10] L.D. Faddeev and L.A. Takhtadjyan Hamiltonian approach to soliton theory, Nauka, Moscow, 1986 (in Russian).
• [11] R.G. Newton: Scattering Theory of Waves and Particles, 2nd ed., Dover Publications, Paperback, 2002.
• [12] R.G. Newton: Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, 1990.
• [13] S.P. Novikov (Ed.): Theory of solitons, Moscow, Nauka Publ., 1980, (in Russian).
• [14] Yu.M. Berezansky: Eigenfunctions expansions related with selfadjoint operators, Nauk. Dumka Publ, Kiev, 1965 (in Russian).
• [15] F.A. Berezin and M.A. Shubin: Schrödinger equation, Moscow University Publisher, Moscow, 1983 (in Russian).
• [16] A.L. Bukhgeim: Volterra equations and inverse problems, Nauka, Moscow, 1983, (in Russian).
• [17] V.B. Matveev and M.I. Salle: Darboux-Bäcklund transformations and applications, Springer, NY, 1993.
• [18] L.P. Nizhnik: Inverse scattering problems for hyperbolic equations, Nauk. Dumka Publ., Kiev, 1991 (in Russian).
• [19] L.P. Nizhnik and M.D. Pochynaiko: “The integration of a spatially two-dimensional Schrödinger equation by the inverse problem method”, Func. Anal. and Appl., Vol. 16(1), (1982), pp. 80–82 (in Russian).
• [20] I.C. Gokhberg and M.G. Krein: Theory of Volterna operators in Hilbert spaces and its applications, Nauka, Moscow, 1967 (in Russian).
• [21] Ya.V. Mykytiuk: “Factorization of Fredholmian operators”, Mathematical Studii, Proceedings of Lviv Mathematical Society, Vol. 20(2), (2003), pp. 185–199 (in Ukrainian).
• [22] A.M. Samoilenko, Y.A. Prykarpatsky and V.G. Samoylenko: “The structure of Darboux-type binary transformations and their applications in soliton theory”, Ukr. Mat. Zhurnal, Vol. 55(12), (2003), pp. 1704–1723 (in Ukrainian).
• [23] Y.A. Prykarpatsky, A.M. Samoilenko and A.K. Prykarpatsky: De Rham-Hodge theory. A survey of the spectral and differential geometric aspects of the De Rham-Hodge theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems. Part 1, //lanl-arXiv:math-ph/0406062 v 1, 8 April 2004.
• [24] A.K. Prykarpatsky, A.M. Samoilenko and Y.A. Prykarpatsky: “The multidimensional Delsarte transmutation operators, their differential-geometric structure and applications. Part. 1”, Opuscula Mathematica, Vol. 23, (2003), pp. 71–80, /arXiv:math-ph/0403054 v1 29 March 2004.
• [25] J. Golenia, Y.A. Prykarpatsky, A.M. Samoilenko and A.K. Prykarpatsky: “The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory, Part 2.”, Opuscula Mathematica, Vol. 24, (2004), /arXiv: math-ph/0403056 v 1 29 March 2004.
• [26] A.M. Samoilenko and Y.A. Prykarpatsky: Algebraic-analytic aspects of completely integrable dynamical systems and their perturbations, Vol. 41, NAS, Inst. Mathem. Publisher, Kiev, 2002 (in Ukrainian).
• [27] Y.A. Prykarpatsky, A.M. Samoilenko, A.K. Prykarpatsky and V.Hr. Samoylenko: The Delsarte-Darboux type binary transformations and their differenetial-geometric and operator staructure, arXiv: math-ph/0403055 v 1 29 March 2004.
• [28] J.C.C. Nimmo: “Darboux tarnsformations from reductions of the KP-hierarchy”, Preprint of the Dept. of Mathem. at the University of Glasgow, November 8, 2002, p. 11.
• [29] A.K. Prykarpatsky and I.V. Mykytiuk: Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects, Kluwer Acad. Publishers, The Netherlands, 1998.
• [30] C. Godbillon: Geometrie differentielle et mechanique analytique, Paris, Hermann, 1969.
• [31] R. Teleman: Elemente de topologie si varietati diferentiabile, Bucuresti Publ., Romania, 1964.
• [32] G. De Rham: Varietes differentielles, Hermann, Paris, 1955.
• [33] G. De Rham: “Sur la theorie des formes differentielles harmoniques”, Ann. Univ. Grenoble, Vol. 22, (1946), pp. 135–152.
• [34] F. Warner: Foundations of differential manifolds and Lie groups, Academic Press, NY, 1971.
• [35] N. Danford and J.T. Schwartz: Linear operators, Vol. 2, InterSci. Publ., NY, 1963.
• [36] B.N. Datta and D.R. Sarkissian: “Feedback control in distributed parameter gyroscopic systems: a solution of the partial eigenvalue assignment problem”, Mechanical systems and Signal Processing, Vol. 16(1), (2002), pp. 3–17. http://dx.doi.org/10.1006/mssp.2001.1444
• [37] I.M. Gelfand and G.E. Shilov: Generalized functions and actions upon them, 2nd ed., Nauka Publisher, Moscow, 1959 (in Russian).
• [38] S.P. Novikov (Ed.): Theory of solitons, Nauka Publ., Moscow, 1980 (in Russian).
• [39] M.D. Pochynaiko and Yu.M. Sydorenko: “Integrating some (2+1)-dimensional integrable systems by methods of inverse scattering problem and binary Darboux transformations”, Matematychni studii, Vol. 20, (2003), pp. 119–132.
• [40] V.E. Zakharov and A.B. Shabat: “A scheme of integration of nonlinear equations of mathematical physics via the inverse scattering problem”, Part 1, Func. Anal. and it Appl., Vol. 8(3), (1974), pp. 43–53; Part 2, Vol. 13(3), (1979), pp. 13–32 (in Russian).
• [41] B.G. Konopelchenko: “On the integrable equations and degenerate dispersiopn laws in multidimensional soaces”, J. Phys. A: Math. and Gen., Vol. 16, (1983), pp. L311-L316. http://dx.doi.org/10.1088/0305-4470/16/9/006
• [42] V.E. Zakharov: “Integrable systems in multidimensional spaces”, Lect. Notes in Phys., Vol. 153, (1982), pp. 190–216. http://dx.doi.org/10.1007/3-540-11192-1_38
• [43] V.E. Zakharov and S.V. Manakov: “On a generalization of the inverse scattering problem”, Theoret. Mathem. Physics, Vol. 27(3), (1976), pp. 283–287.
• [44] D. Levi, L. Pilloni and P.M. Santini: “Bäcklund transformations for nonlinear evolution equations in (2+1)-dimensions”, Phys. Lett A., Vol. 81(8), (1981), pp. 419–423. http://dx.doi.org/10.1016/0375-9601(81)90401-1
• [45] Liu Wen: Darboux transformations for a Lax integrable systems in 2n-dimensions, arXive:solve-int/9605002 v1 15 May 1996.
• [46] C.H. Gu: Generalized self-dual Yang-Mills flows, explicit solutions and reductions. Acta Applicandae Mathem., Vol. 39, (1995), pp. 349–360. http://dx.doi.org/10.1007/BF00994642

Identyfikatory

DOI

10.2478/BF02475922