Now showing items 1-16 of 16

    • Bi-presymplectic chains of co-rank 1 and related Liouville integrable systems 

      Błaszak, M.; Gürses, M.; Zheltukhin, K. (Institute of Physics Publishing Ltd., 2009)
      Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such chains represent some Liouville integrable systems and the conditions under which there exist related bi-Hamiltonian chains ...
    • Discrete symmetries and nonlocal reductions 

      Gürses, Metin; Pekcan, A.; Zheltukhin, K. (Elsevier, 2020)
      We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
    • Dynamical systems and Poisson structures 

      Gurses, M.; Guseinov, G. Sh.; Zheltukhin, K. (American Institute of Physics, 2009)
      We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in ℝ3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical ...
    • Hamiltonian equations in ℝ3 

      Ay, A.; Gürses, M.; Zheltukhin, K. (AIP Publishing LLC, 2003-08)
      The Hamiltonian formulation of N=3 systems is considered in general. The most general solution of the Jacobi equation in ℝ3 is proposed. The form of the solution is shown to be valid also in the neighborhood of some irregular ...
    • Hydrodynamic type of integrable equations on a segment and a half line 

      Gurses, M.; Habibullin, I.; Zheltukhin, K. (American Institute of Physics, 2008)
      The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions ...
    • Integrable boundary value problems for elliptic Toda lattice in a disk 

      Gürses M.; Habibulin, I.; Zheltukhin, K. (American Institute of Physics, 2007)
      The concept of integrable boundary value problems for soliton equations on ℝ and ℝ+ is extended to regions enclosed by smooth curves. Classes of integrable boundary conditions in a disk for the Toda lattice and its reductions ...
    • Nonlocal hydrodynamic type of equations 

      Gürses, Metin; Pekcan, A.; Zheltukhin, K. (Elsevier, 2020-03-01)
      We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of ...
    • On a class of Darboux-integrable semidiscrete equations 

      Zheltukhin, K.; Zheltukhina, N.; Bilen, E. (Springer, 2017)
      We consider a classification problem for Darboux-integrable hyperbolic semidiscrete equations. In particular, we obtain a complete description for a special class of equations admitting four-dimensional characteristic ...
    • On a transformation between hierarchies of integrable equations 

      Gürses, M.; Zheltukhin, K. (Elsevier BV * North-Holland, 2006)
      A transformation between a hierarchy of integrable equations arising from the standard R-matrix construction on the algebra of differential operators and a hierarchy of integrable equations arising from a deformation of ...
    • On existence of an x-integral for a semi-discrete chain of hyperbolic type 

      Zheltukhin, K.; Zheltukhina, Natalya (IOP, 2016)
      A class of semi-discrete chains of the form t1x = f (x,t,t1,tx) is considered. For the given chains easily verifiable conditions for existence of x-integral of minimal order 4 are obtained.
    • On the discretization of darboux integrable systems 

      Zheltukhin, K.; Zheltukhina, Natalya (Taylor and Francis, 2020)
      We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
    • On the discretization of Laine equations 

      Zheltukhin, K.; Zheltukhina, Natalya (Taylor and Francis, 2018)
      We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such ...
    • On the integrability of a class of Monge-Ampère equations 

      Brunelli J.C.; Gürses M.; Zheltukhin, K. (World Scientific Publishing Co. Pte. Ltd., 2001)
      We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge-Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless ...
    • Recursion operator and dispersionless rational Lax representation 

      Zheltukhin, K. (2002)
      We consider equations arising from dispersionless rational Lax representations. A general method to construct recursion operators for such equations is given. Several examples are given, including a degenerate bi-Hamiltonian ...
    • Recursion operators of some equations of hydrodynamic type 

      Gürses M.; Zheltukhin, K. (A I P Publishing LLC, 2001)
      We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give several examples for N = 2 and N = 3 containing the equations of ...
    • Semi-discrete hyperbolic equations admitting five dimensional characteristic x-ring 

      Zheltukhin, K.; Zheltukhina, N. (Taylor and Francis Ltd., 2016)
      The necessary and sufficient conditions for a hyperbolic semi-discrete equation to have five dimensional characteristic x-ring are derived. For any given chain, the derived conditions are easily verifiable by straightforward ...