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Abstrakty
The structure properties of multidimensional Delsarte-Darboux transmutation operators in parametric functional spaces are studied by means of differential-geometric and topological tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutative differential operator expressions related via a Delsarte-Darboux transformation and having a lot of applications in soliton theory.
Czasopismo
Rocznik
Tom
Strony
137--150
Opis fizyczny
Bibliogr. 22 poz.
Twórcy
autor
autor
- AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland, yarchyk@gmail.com
Bibliografia
- [1] Delsarte J., Sur certaines transformations fonctionelles relative aux equations linearines aux derives partielles du second ordre, C.R. Acad. Sci. Paris 206 (1938), 178-182.
- [2] Delsarte J., Lions J., Transmutations d'operateurs differentiles dans le domain complexe, Comment. Math. Helv. 52 (1957), 113-128.
- [3] Berezin F. A., Shubin M. A., The Schrodinger equation, Moscow, Nauka Publ.. 1983 (in Russian).
- [4] Berezansky Yu.M., Eigenfunctions expansions with respect to self-adjoint differential operators, Kiev, Nauk.Dumka Publ., 1965 (in Russian).
- [5] Berezin F. A., Shubin M. A., Schrodinger equation, Moscow, the Moscow University Publisher, 1983 (in Russian).
- [6] Marchenko V. A., Spectral theory of Sturm-Liouville operators, Kiev, Nauk. Dumka Publ., 1972 (in Russian).
- [7] Levitan B.M., Sargsian I.S., Sturm-Liouville and Dirac operators, Moscow Nauka Publ., 1988 (in Russian).
- [81 Matveev V.B., Salle M.I., Drboux-Backlund transformations and applications, NY: Springer, 1993.
- [9] Novikov S.P. (Ed.), Soliton theory, Moscow. Nauka, 1980, 319.
- [10] Ablowitz M.J., Segur H., Solitons and the invesre scattering transform, SIAM, Philadelphia, USA, 1981.
- [11] Nimmo J.C.C., Darboux transformations from reductions of the KP-hierarchy, Preprint of the Department of Mathematics at the University of Glasgow. November, 8, 2002, 11 p.
- [12] Golenia J., Prykarpatsky Y. A., Samoilenko A.M., Prykarpatsky A. K., The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2, Opuscula Mathematica, 24 (2004) 1, 74-83.
- [13] Samoilenko A.M., Prykarpatsky Ya. A., Algebraic-analytic aspects of completely integrable dynamical systems and their perturbations. Proceedings of Institute of Mathematics of NAS of Ukraina. - Kyiv: Institute of Mathematics (41) 2002, 236.
- [14] Prykarpatsky, A. K., Mykytiuk, I. V., Algebraic integrability of dynamical systems on manifolds, Kluwer Academic Publisher, the Netherlands, 1998.
- [15] Nizhnik L. P., Integrability of nonlinear multidimensional equations with inverse scattering transform approach, Doklady of AN USSR. 254 (1980)2 332-335.
- [16] Nizhnik L.P., Inverse scattering problems for hyperbolic equations, Kyiv, Nauk. dumka, 1991, 232 p.
- [17] Manakov S.V., Inverse scattering transform approach and two-dimensional evolution equations, Uspekhi mat. nauk. 31 (1976) 5, 245-246.
- [18] Godbillon K., Differential geometry and analitical mechanics, Moscow, Mir, 1973, 188 p.
- [19] Cartan A., Differential calculus. Differential forms, Moscow, Mir, 1971, 392 p.
- [20] Hentosh O.E., Prytula M. M., Prykarpatsky A. K., Differential -geometric and algebraic foundations of studying integrable nonlinear dynamical systems on functional manifolds, Lviv University Publisher, Ukraine, 2005 (in Ukrainian).
- [21] Prykarpatsky A. K., Samoilenko A.M., Prykarpatsky Y. A., The multidimensional Delsarte transmutation operators, their differential-geometric structure and applications. Part 1., Opuscula Mathematica, 23 (2003), 71-80
- [22] Prykarpatsky Y. A., Samoilenko A.M., Samolyenko V. G., The structure of Darboux-type binary transformations and their applications in soliton theory, Ukr. Mat. Zhurnal, 55 (2003)12, 1704-1719 (in Ukrainian).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGH4-0006-0010