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Slezská univerzita v Opavě
Matematický ústav v Opavě

Doc. RNDr. Artur Sergyeyev, Ph.D.
Position Associate Professor
Address Mathematical Institute
Silesian University in Opava
Na Rybnicku 1
746 01 Opava, Czech Republic
Phone +420 553 684 685
Fax +420 553 684 680
E-mail

Curriculum

Academic Degrees

Professional Experience

Awards and Honors

  • The Prize of the Rector of Silesian University in Opava, 2002
  • The winner of Second All-Ukrainian Olympiad in Applied Mathematics among 2nd - 5th year students, 1995

Research Interests

  • Recursion operators, formal symmetries, local and nonlocal generalized symmetries of partial differential equations, and their applications
  • Conditional integrability and exact solutions of partial differential equations
  • Integrability tests, classification and construction of integrable systems
  • Exactly and quasiexactly solvable models of quantum mechanics
  • Supersymmetries and parasupersymmetries in quantum mechanics

List of publications

Please note: A.G. Sergheyev = A.G. Sergeyev = A.G. Sergyeyev

A. Refereed Journal Publications

  1. A. Sergyeyev and B. M. Szablikowski, Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems, Phys. Lett. A, 2008, 372, 7016-7023 (arXiv version).
  2. A. Sergyeyev and P. Krtouš, Complete set of commuting symmetry operators for the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetimes, Phys. Rev. D, 2008, 77, paper 044033 (arXiv version)
  3. A. Sergyeyev  and M. Blaszak, Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems, J. Phys. A: Math. Theor., 2008, 41, paper 10525 (arXiv version)
  4. A. Sergyeyev, Exact solvability of superintegrable Benenti systems, J. Math. Phys., 2007, 48, no.5, paper 052114 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  5. M. Blaszak, A. Sergyeyev, Natural coordinates for a class of Benenti systems,  Phys. Lett. A, 2007, 365, no.1-2, 28-33 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  6. A. Sergyeyev, D. Demskoi, Sasa-Satsuma (complex modified Korteweg—de Vries II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries, and more, J. Math. Phys., 2007, 48, no.4, paper 042702 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  7. A. Sergyeyev, Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility, SIGMA, 2007, 3, paper 062 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  8. A. Sergyeyev, Zero curvature representation for a new fifth-order integrable system (in Russian, English translation to appear in J. Math. Sci.), Fund. Prikl. Mat., 2006, 12, no.7, 227-229 Preprint version (fulltext in English in arxiv.org a.k.a. LANL preprint archive)
  9. A. Sergyeyev, A strange recursion operator demystified, J. Phys. A: Math. Gen., 2005, 38, no.15, L257-L262 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  10. A. Sergyeyev, Why nonlocal recursion operators produce local symmetries: new results and applications, J. Phys. A: Math. Gen., 2005, 38, no.15, 3397-3407 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  11. M. Blaszak, A. Sergyeyev, Maximal superintegrability of Benenti systems, J. Phys. A: Math. Gen., 2005, 38, no.1, L1-L5 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  12. A. Sergyeyev, Towards classification of conditionally integrable evolution systems in (1+1) dimensions, Fund. Prikl. Mat., 2004, 10,  no. 1, 243-253 (in Russian), Preprint version (fulltext in English in arxiv.org a.k.a. LANL preprint archive)
  13. A. Sergyeyev, A simple way of making a Hamiltonian system into a bi-Hamiltonian one, Acta Appl. Math., 2004, 83, no.1-2, 183-197 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  14. A. Sergyeyev, Locality of symmetries generated by nonhereditary, inhomogeneous, and time-dependent recursion operators: a new application for formal symmetries, Acta Appl. Math., 2004, 83, no.1-2, 95-109  Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  15. M. Marvan, A. Sergyeyev, Recursion operator for the stationary Nizhnik--Veselov--Novikov equation, J. Phys. A: Math. Gen., 2003, 36, no.5, L87-L92 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  16. A. Sergyeyev and J.A. Sanders, A remark on nonlocal symmetries for the Calogero--Degasperis--Ibragimov--Shabat equation, J. Nonlin. Math. Phys., 2003, 10, no. 1, 78-85 ArXiv copy (fulltext in arxiv.org a.k.a. LANL preprint archive)
  17. A. Sergyeyev, On sufficient conditions of locality for hierarchies of symmetries of evolution systems, Rep. Math. Phys., 2002, 50, no.3, 307-314 Preprint version (fulltext in gzipped PostScript format)
  18. A. Sergyeyev, Constructing conditionally integrable evolution systems in (1+1) dimensions: a generalization of invariant modules approach, J. Phys. A: Math. Gen., 2002, 35, 7653-7660 Preprint version (fulltext in gzipped PostScript format)
  19. A. Sergyeyev, On homogeneous symmetries for evolution systems with constraints, Rendiconti del Circolo Matematico di Palermo, 2002, Serie II, No. 69, 219-231. Preprint version (fulltext in gzipped PostScript format)
  20. A. Sergyeyev, Symmetries and integrability: Bakirov system revisited, J. Phys. A: Math. Gen.,2001, 34, no.23, 4983-4990.
  21. A. Sergyeyev, On symmetries of KdV-like evolution equations, Rep. Math. Phys.,1999, 44, no.1/2, 183-190 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive).
  22. A. Sergheyev, Generalized symmetries of partial differential equations and quasiexact solvability, Rep. Math. Phys., 1998, 41, no.3, 279-286 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive).
  23. A.G. Sergheyev, A relativistic Coulomb problem for the modified Stueckelberg equation, Ukr. J. Phys., 1997, 42, no.10, 1171-1174 Preprint version

B. Articles in Refereed Proceedings

  1. A. Sergyeyev, The structure of cosymmetries and a simple proof of locality for hierarchies of symmetries of odd order evolution systems, Proc. Fifth Int. Conf. ``Symmetry in Nonlinear Mathematical Physics", published in Proceedings of Institute of Mathematics of NAS of Ukraine, 2004,  50, Part 1, p.238-245.
  2. A. Sergyeyev, On a class of inhomogeneous extensions for integrable evolution systems, Proc. 8th Int. Conf. on Diff.Geom. and its Appl., 2001, p.243-252 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  3. A. Sergyeyev and J.A. Sanders, The Complete Set of Generalized Symmetries for the Calogero--Degasperis--Ibragimov--Shabat Equation, Proc. 4th Int. Conf. "Symmetry in Nonlinear Mathematical Physics", published in Proceedings of Institute of Mathematics of NAS of Ukraine,2002, 43, Part 1, p.209-214.
  4. A. Sergyeyev, On local time-dependent symmetries of integrable evolution equations, Proc. 3rd Int. Conference "Symmetry in Nonlinear Mathematical Physics" (Kyiv, July 13-18, 1999), published in Proceedings of Institute of Mathematics of NAS of Ukraine, 2000, 30, Part 1, p.196-203.
  5. A. Sergyeyev, Nonlocal symmetries and formal symmetries for evolution equations, Group and Analytic Methods in Mathematical Physics, Proceedings of Institute of Mathematics of NAS of Ukraine, 2001, 35, p.227-236.
  6. A. Sergyeyev, On recursion operators and nonlocal symmetries of evolution equations, Proc. Sem. Diff. Geom., D. Krupka ed., Mathematical Publications, Silesian University in Opava, Opava, Czech Republic, 2000, 2, p.159-173 Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  7. A. Sergyeyev, On time-dependent symmetries and formal symmetries of evolution equations, in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta eds., Singapore, World Scientific, 1999, p.303-308.
  8. A.G. Sergyeyev, On the class of equations for the massless particles that are invariant under Poincaré parasupergroup, in Proc. 2nd All-Ukrainian Conference of young scientists (Kyiv, 16-18 May 1995), Kyiv, 1995, p.41-46 (in Ukrainian).

C. Other Works

  1. A. Sergyeyev, Time dependence and (non)commutativity of symmetries of evolution equations, in "Noncommutative Structures in Mathematics in Physics" (Proc. NATO ARW), S. Duplij and J. Wess eds., Kluwer, 2001, p.379-390. Preprint version (fulltext in arxiv.org a.k.a. LANL preprint archive)
  2. A.G. Sergyeyev, Higher Symmetries and Parasupersymmetries of Evolution Equations, Ph.D. thesis, Kyiv, Institute of Mathematics of NAS of Ukraine, 2000, 130 p. (in Ukrainian).
  3. A.G. Sergyeyev, On the structure of generalized symmetries for nonlinear evolution equations, Visnyk Kyivs'kogo Universytetu (Bulletin of the University of Kiev), Series: Physical and mathematical Sciences, 1999, no.3, p.61-70 (in Ukrainian).
  4. A.G. Sergheyev, On time-dependent symmetries of evolution equations, Symmetry and Analytic Methods in Mathematical Physics, Proceedings of Institute of Mathematics of NAS of Ukraine, 1998, 19, p. 216-220.
  5. A.G. Sergeyev, On parasupersymmetries in a relativistic Coulomb problem for the modified Stueckelberg equation, in Proc. 2nd Int. Conference "Symmetry in Nonlinear Mathematical Physics" (Kyiv, July 7-13, 1997), Kyiv, 1997, 2, p.331-335.
  6. O.F. Kalaida, A.G. Sergyeyev, O.V. Galkin, New variant of matrix algorithms of numerical integration for regular, weakly singular, and singular integrals, Visnyk Kyivs'kogo Universytetu (Bulletin of the University of Kiev), Series: Physical and Mathematical Sciences, 1993, no.3, p.18-22 (in Ukrainian).

Fellowships and Grants

  • Team Grant MSM4781305904 of the Ministry of Education, Youth and Sports of Czech Republic
  • The personal development project No. 254/b for the year 2004 of the Ministry of Education, Youth and Sports of Czech Republic
  • Internal Grant IGS 1/2004 of Silesian University in Opava
  • Team Grant No. 201/04/0538 of the Czech Grant Agency (Principal Investigator: Michal Marvan)
  • The Jacob Blaustein Postdoctoral Fellowship, The Jacob Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Israel (April 29, 2003 - May 10, 2004)
  • Postdoctoral Fellowship at Graduiertenkolleg "Geometrie und Nichtlineare Analysis", Institut für Mathematik, Humboldt-Universität zu Berlin (April 1 - December 31, 2001)
  • Team Grant No. 201/00/0724 of the Czech Grant Agency (Principal Investigator: Demeter Krupka)
  • Team Grants MSM:J10/98:192400002 and VS 96003 "Global Analysis" of the Ministry of Education, Youth and Sports of Czech Republic
  • ISSEP (Soros Students Program) Personal Grant GSU052172 (1995)
  • Personal Studentship from Kyiv National Taras Shevchenko University (1994--1996)

Main Scientific Results

  • Proof of maximal superintegrability for a large class of Benenti systems (natural Hamiltonian dynamical systems admitting a special conformal Killing tensor) -- joint work with M. Blaszak (see [A.3])
  • For a large class of (1+1)-dimensional evolution systems, new easily verifiable sufficient conditions ensuring the locality of symmetries generated via the repeated application of recursion operator (see [A.2, A.6, B.1]) or repeated commutation with master symmetry (see [A.9, B.1]) were obtained. In particular, using these results we proved the Maltsev--Novikov conjecture on weak nonlocality of higher recursion operators, Hamiltonian and symplectic structures of integrable systems in (1+1) dimensions (see [A.2]).
  • A new description of Hamiltonian operators compatible with a given nondegenerate Hamiltonian operator, applicable for both finite- and infinite-dimensional case (see [A.5])
  • The complete classification of (1+1)-dimensional evolution systems admitting a generalized conditional symmetry (GCS) of prescribed form, including (unlike the known results of Svirshchevskii, Kamran, Milson, and Olver and Doyle) the case of explicitly time-dependent GCS (see [A.4, A.10]).
  • All local generalized symmetries of the Bakirov system, including those explicitly dependent on independent variables, were found. In particular, it was shown that this system has only one non-Lie-point local generalized symmetry. This result, generalizing earlier work of Beukers, Sanders, and Wang, completes the refutation of the popular conjecture that the existence of one non-Lie point generalized symmetry for a system of PDEs implies the existence of infinitely many such symmetries (see [A.12]).
  • An explicit formula for the leading term of commutator of two generalized symmetries of sufficiently high order was obtained (see [A.11]). This formula is valid for a large class of (1+1)-dimensional evolution systems with constraints. It possesses a number of useful applications and provides the complete solution of A.M. Vinogradov's problem of 'evaluation from the top' for the symmetry algebras of these systems.
  • The first example of parasupersymmetric relativistic quantum-mechanical system with non-oscillator-like interaction was found. The complete set of eigenvalues and eigenvectors belonging to the discrete energy spectrum for this model was found (see [C.5]).

Talks Given at the Following Conferences

  • 10th International Conference on Differential Geometry and its Applications (August 27-31, 2007, Olomouc, Czech Republic)
  • Seventh International Conference "Symmetry in Nonlinear Mathematical Physics" (June 24-30, 2007, Kyiv, Ukraine)
  • 16th International Colloquium "Quantum Groups and Integrable Systems" (June 14-16, 2007, Prague, Czech Republic)
  • The 27th Winter School "Geometry and Physics" (January 13-20, 2007, Srni, Czech Republic)
  • Eduard Čech Centre Meeting (November 10-11, 2006, Telč, Czech Republic)
  • 15th International Colloquium "Quantum Groups and Integrable Systems" (June 15-17, 2006, Prague, Czech Republic)
  • Sixth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 20-26, 2005, Kyiv, Ukraine)
  • 14th International Colloquium "Quantum Groups and Integrable Systems" (June 16-18, 2005, Prague, Czech Republic)
  • The 25th Winter School "Geometry and Physics" (January 15-22, 2005, Srni, Czech Republic)
  • 10th International Conference on Modern Group Analysis (MOGRAN X) (October 24-31, 2004, Larnaca, Cyprus)
  • 9th International Conference on Differential Geometry and its Applications (August 30-September 3, 2004, Prague, Czech Republic)
  • 11th International Conference "Symmetry Methods in Physics" (June 21-24, 2004, Prague, Czech Republic)
  • 13th International Colloquium "Quantum Groups and Integrable Systems" (June 17-19, 2004, Prague, Czech Republic)
  • Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv, Ukraine)
  • Fifth Negev Workshop on Applied Mathematics (June 17-21, 2003, Sede Boqer, Israel)
  • The 23rd Winter School "Geometry and Physics" (January 18-25, 2003, Srni, Czech Republic)
  • 11th International Colloquium "Quantum Groups and Integrable Systems" (June 20-22, 2002, Prague, Czech Republic)
  • The 22nd Winter School "Geometry and Physics" (January 12-19, 2002, Srni, Czech Republic)
  • 8th International Conference on Differential Geometry and its Applications (August 27-31, 2001, Opava, Czech Republic)
  • Banach Center Workshop "Multi-Hamiltonian Structures: Geometric and Algebraic Aspects" (August 9-17, 2001, Bedlewo, Poland)
  • Fourth International Conference "Symmetry in Nonlinear Mathematical Physics" (July 9-15, 2001, Kyiv, Ukraine)
  • The 21st Winter School "Geometry and Physics" (January 13-20, 2001, Srní, Czech Republic)
  • NATO Advanced Research Workshop "Noncommutative Structures in Mathematics and Physics" (September 24-28, 2000, Kyiv, Ukraine)
  • Third International Conference "Symmetry in Nonlinear Mathematical Physics" (July 13-18, 1999, Kyiv, Ukraine)
  • XXXI Symposium on Mathematical Physics (May 18-21, 1999, Torun, Poland)
  • XXX Jubilee Symposium on Mathematical Physics (May 26-30, 1998, Torun, Poland)
  • Seventh International Academician's M. Krawtchouk Conference (May 14-16, 1998, Kyiv, Ukraine)
  • Second International Conference "Symmetry in Nonlinear Mathematical Physics" (July 7-13, 1997, Kyiv, Ukraine)
  • Sixth International Academician's M. Krawtchouk Conference (May 15-17, 1997, Kyiv, Ukraine)
  • Second All-Ukrainian Conference of Young Scientists (May 16-18, 1995, Kyiv, Ukraine)

Poster presented at

  •  Seminar Sophus Lie (December 6-8, 2001, Berlin, Germany)

Participation at the Schools for Young Scientists

  • Workshop Recent Trends in Mathematics (Schlosshotel Klink, Germany, November 23-25, 2001)
  • Advanced Course Variational Sequences and Bicomplexes (Opava, Czech Republic, August 20-24, 2001)
  • The 5th CIME Session Direct and Inverse Methods in Solving Nonlinear Evolution Equations (Grand Hotel San Michele, Cetraro, Italy, September 6-11, 1999)

Short Scientific Visits

  • Institute of Physics, Adam Mickiewicz University (Poznan, Poland, May 20-June 9, 2007, host: Prof. M. Blaszak)
  • Centre de Recherches Mathématiques, Université de Montréal, (Montréal, Québec, Canada, November 15-December 11, 2004, host: Pavel Winternitz)
  • Universität Paderborn (Paderborn, Germany, December 5-6, 2001, host: Benno Fuchssteiner)
  • Vrije Universiteit van Amsterdam (Amsterdam, The Netherlands, February 18 - March 4, 2001, host: Jan A. Sanders)
  • Mathematical Institute of Silesian University in Opava (Opava, Czech Republic, January 10-21, 2000, host: Michal Marvan)

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