Artur Sergyeyev

Publications (5)0 Total impact

• Source

  Available from: ArXiv

  Article: A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

  Maciej Blaszak, Artur Sergyeyev
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  ABSTRACT: Using a (1,1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we show that these new systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreover, are weakly nonlinear and semi-Hamiltonian, so their general solution can be obtained by means of the generalized hodograph method of Tsarev.
  04/2008;
• Source

  Available from: ArXiv

  Article: Generalized St\"ackel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems

  Artur Sergyeyev, Maciej Blaszak
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  ABSTRACT: We present a multiparameter generalization of the St\"ackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized St\"ackel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized St\"ackel transform.
  07/2007;
• Source

  Available from: arxiv.org

  Article: Reciprocal transformations for Stackel-related Liouville integrable systems

  Maciej Blaszak, Artur Sergyeyev
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  ABSTRACT: We consider the St\"ackel transform, also known as the coupling-constant metamorphosis, which under certain conditions turns a Hamiltonian dynamical system into another such system and preserves the Liouville integrability. We show that the corresponding transformation for the equations of motion is nothing but the reciprocal transformation of a special form and we investigate the properties of this transformation. This result is further applied for the study of the $k$-hole deformations of the Benenti systems or more general seed systems.
  08/2006;
• Source

  Available from: ArXiv

  Article: Natural coordinates for a class of Benenti systems

  Maciej Błaszak, Artur Sergyeyev
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  ABSTRACT: We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.
  Physics Letters A. 05/2006;
• Article: A coordinate-free construction of conservation laws and reciprocal transformations for a class of integrable hydrodynamic-type systems

  Maciej Błaszak, Artur Sergyeyev
  [show abstract] [hide abstract]
  ABSTRACT: Using a (1, 1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we show that these new systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreover, are weakly nonlinear and semi-Hamiltonian, so their general solution can be obtained by means of the generalized hodograph method of Tsarev.
  Reports on Mathematical Physics.