Kira V. Khmelnytskaya

Autonomous University of Queretaro, Ciudad Queretaro, Querétaro, Mexico

Are you Kira V. Khmelnytskaya?

Claim your profile

Publications (23)19.56 Total impact

  • Haret C. Rosu, Kira V. Khmelnytskaya
    [Show abstract] [Hide abstract]
    ABSTRACT: It is known that the barotropic FRW system of differential equations can be reduced to simple harmonic oscillator (HO) differential equations in the conformal time variable. This is due to the fact that the Hubble rate parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. In previous works, we have used this mathematical result to set the barotropic HO equations in the nonrelativistic supersymmetric approach by factorizing them. If a constant additive parameter, denoted by S, is added to the common Riccati solution of these supersymmetric partner cosmologies one obtains inhomogeneous barotropic cosmologies with periodic singularities in their spatial curvature indices that are counterparts of the non-shifted supersymmetric partners. The zero-mode solutions of these cyclic singular cosmologies are reviewed here as a function of real and imaginary shift parameter. We also notice the modulated zero modes obtained by using the general Riccati solution and comment on their cosmological application.
    12/2013;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.
    Journal of Mathematical Analysis and Applications 03/2013; 399(1):191-212. · 1.05 Impact Factor
  • Haret C. Rosu, Kira V. Khmelnytskaya
    [Show abstract] [Hide abstract]
    ABSTRACT: It is known that the barotropic FRW system of differential equations for zero cosmological constant can be reduced to simple harmonic oscillator (HO) differential equations in the conformal time variable. This is due to the fact that the Hubble rate parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. In previous works, we have used this mathematical result to set the barotropic HO equations in the nonrelativistic supersymmetric approach by factorizing them. If a constant additive parameter, denoted by S, is added to the common Riccati solution of these supersymmetric partner cosmologies one obtains inhomogeneous barotropic cosmologies with periodic singularities in their spatial curvature indices that are counterparts of the non-shifted supersymmetric partners. The zero-mode solutions of these cyclic singular cosmologies are reviewed here as a function of real and imaginary shift parameter. We also notice the modulated zero modes obtained by using the general Riccati solution and comment on their cosmological application.
    Modern Physics Letters A 01/2013; 28(3):40017-. · 1.11 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A general solution of the fourth-order Sturm–Liouville equation is presented in the form of a spectral parameter power series (SPPS). The uniform convergence of the series is proved and the coefficients of the series are calculated explicitly through a recursive intergration procedure. Based on the SPPS representation characteristic equations for spectral problems arising in mechanics and elasticity theory are obtained and it is shown that the spectral problems reduce to computation of zeros of corresponding analytic functions of the spectral parameter given by their Taylor series expansions. This leads to a simple and efficient numerical method for solving the spectral problems for fourth-order Sturm–Liouville equations. Several examples of application are discussed.
    Applied Mathematics and Computation 12/2012; 219(8):3610–3624. · 1.35 Impact Factor
  • Source
    K. V. Khmelnytskaya, V. V. Kravchenko, H. C. Rosu
    [Show abstract] [Hide abstract]
    ABSTRACT: In the present review we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm-Liouville equations with variable coefficients. We consider Sturm-Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet-Bloch solutions; quantum-mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov-Shabat system. In all these cases we obtain a characteristic equation of the problem which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples which show that at present the SPPS method is the easiest in the implementation, the most accurate and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and for practical purposes constitutes a powerful numerical method but most important it offers additional insight into the spectral and transmission problems.
    Mathematical Methods in the Applied Sciences 12/2011; · 0.78 Impact Factor
  • H. C. Rosu, K. V. Khmelnytskaya
    [Show abstract] [Hide abstract]
    ABSTRACT: We determine the kind of parametric oscillators that are generated in the usual factorization procedure of second-order linear differential equations when one introduces a constant shift of the Riccati solution of the classical harmonic oscillator. The mathematical results show that some of these oscillators could be of physical nature. We give the solutions of the obtained second-order differential equations and the values of the shift parameter providing strictly periodic and antiperiodic solutions. We also notice that this simple problem presents parity-time (PT) symmetry. Possible applications are mentioned.
    Physics Letters A 01/2011; 375(40):3491-3495. · 1.63 Impact Factor
  • Source
    Haret C. Rosu, Kira V. Khmelnytskaya
    [Show abstract] [Hide abstract]
    ABSTRACT: In the case of barotropic FRW cosmologies, the Hubble parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. We consider these cosmologies in the framework of nonrelativistic supersymmetry that has been effective in the area of supersymmetric quantum mechanics. Recalling that Faraoni [Amer. J. Phys. 67 (1999), 732-734] showed how to reduce the barotropic FRW system of differential equations to simple harmonic oscillator differential equations, we set the latter equations in the supersymmetric approach and divide their solutions into two classes of 'bosonic' (nonsingular) and 'fermionic' (singular) cosmological zero-mode solutions. The fermionic equations can be considered as representing cosmologies of Stephani type, i.e., inhomogeneous and curvature-changing in the conformal time. We next apply the so-called shifted Riccati procedure by introducing a constant additive parameter, denoted by S, in the common Riccati solution of these supersymmetric partner cosmologies. This leads to barotropic Stephani cosmologies with periodic singularities in their spatial curvature indices that we call U and V cosmologies, the first being of bosonic type and the latter of fermionic type. We solve completely these cyclic singular cosmologies at the level of their zero modes showing that an acceptable shift parameter should be purely imaginary, which in turn introduces a parity-time (PT) property of the partner curvature indices.
    Symmetry Integrability and Geometry-methods and Applications - SYMMETRY INTEGR GEOM. 12/2010;
  • Source
    H. C. Rosu, K. V. Khmelnytskaya
    [Show abstract] [Hide abstract]
    ABSTRACT: We determine the kind of parametric oscillators that are generated in the usual factorization procedure of second-order linear differential equations when one introduces a constant shift of the Riccati solution of the classical harmonic oscillator. The mathematical results show that some of these oscillators could be of physical nature. We give the solutions of the obtained second-order differential equations and the values of the shift parameter providing strictly periodic and antiperiodic solutions. We also notice that this simple problem presents parity-time (PT) symmetry. Possible applications are mentioned
    08/2010;
  • Source
    K. V. Khmelnytskaya, H. C. Rosu
    [Show abstract] [Hide abstract]
    ABSTRACT: We provide the representation of quasi-periodic solutions of periodic Dirac equations in terms of the spectral parameter power series (SPPS) recently introduced by V.V. Kravchenko (2008,2009). We also give the SPPS form of the Dirac Hill discriminant under the Darboux nodeless transformation using the SPPS form of the discriminant and apply the results to one of Razavy's quasi-exactly solvable periodic potentials
    06/2010;
  • Source
    K.V. Khmelnytskaya, H.C. Rosu
    [Show abstract] [Hide abstract]
    ABSTRACT: We establish a series representation of the Hill discriminant based on the spectral parameter power series (SPPS) recently introduced by Kravchenko. We also show the invariance of the Hill discriminant under a Darboux transformation and employing the Mathieu case the feasibility of this type of series for numerical calculations of the eigenspectrum.
    Annals of Physics 02/2010; · 3.32 Impact Factor
  • Source
    K. V. Khmelnytskaya, H. C. Rosu, A. Gonzalez
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetrically-coupled second-order differential equations. We solve analytically these parametric periodic problems along the positive real axis. Next, the analytically solved model is used as a case study for a powerful numerical approach that is employed here for the first time in the investigation of the energy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and implements the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill discriminant based on Kravchenko's series Comment: 15 pages, 4 eps figures, a few minor changes, version accepted to Ann. Phys
    Annals of Physics 10/2009; · 3.32 Impact Factor
  • Kira V. Khmelnytskaya, Tetyana V. Torchynska
    [Show abstract] [Hide abstract]
    ABSTRACT: A method for reconstructing symmetric potentials of Schrödinger operators from a finite set of eigenvalues is presented. The method combines the approach developed by Rundell and Coworkers (SIAM Monographs on Mathematical Modeling and Computation. SIAM: Philadelphia, PA; (1997)) for solving inverse Sturm–Liouville problems with a recent result by Kravchenko (Complex Variables and Elliptic Equations 2008; 53(8):775–789) giving accurate solutions of direct problems.Our construction allows one to recover the potential in situations of great importance in studying nanostructures including quantum dots when only a very limited number of eigenvalues (3–4) obtained experimentally is available. Copyright © 2009 John Wiley & Sons, Ltd.
    Mathematical Methods in the Applied Sciences 08/2009; 33(4):469 - 472. · 0.78 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The reflection and transmission of finite inhomogeneous layers are considered. A new accurate and efficient method for calculation of the reflectance and the transmittance is proposed. The method is based on a recently found new representation for solutions of Sturm–Liouville equations.
    Journal of Optics A Pure and Applied Optics 03/2009; 11(6):065707. · 1.92 Impact Factor
  • Source
    Kira V. Khmelnytskaya, Vladislav V. Kravchenko
    [Show abstract] [Hide abstract]
    ABSTRACT: We give an overview of recent advances in analysis of equations of electrodynamics with the aid of biquaternionic technique. We discuss both models with constant and variable coefficients, integral representations of solutions, a numerical method based on biquaternionic fundamental solutions for solving standard electromagnetic scattering problems, relations between different operators of mathematical physics including the Schrodinger, the Maxwell system, the conductivity equation and others leading to a deeper understanding of physics and mathematical properties of the equations. Comment: 1 figure
    02/2009;
  • Source
    K. V. Khmelnytskaya, H. C. Rosu
    [Show abstract] [Hide abstract]
    ABSTRACT: In the context of bilayer graphene we use the simple gauge model of Jackiw and Pi to construct its numerical solutions in powers of the bias potential V according to a general scheme due to Kravchenko. Next, using this numerical solutions, we develop the Ermakov-Lewis approach for the same model. This leads us to numerical calculations of the Lewis-Riesenfeld phases that could be of forthcoming experimental interest for bilayer graphene. We also present a generalization of the Ioffe-Korsch nonlinear Darboux transformation
    Journal of Physics A Mathematical and Theoretical 01/2009; · 1.77 Impact Factor
  • K. Khmelnytskaya, V. Kravchenko, H. Oviedo
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider the static Maxwell system with an axially symmetric dielectric permittivity and construct complete systems of its solutions which can be used for analytic and numerical solution of corresponding boundary value problems.
    Mathematical Methods in Electromagnetic Theory, 2008. MMET 2008. 12th International Conference on; 01/2008
  • Source
    Kira V. Khmelnytskaya, Vladislav V. Kravchenko
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem.
    Journal of Physics A Mathematical and Theoretical 06/2007; · 1.77 Impact Factor
  • Source
    Kira V. Khmelnytskaya, Vladislav V. Kravchenko, Hector Oviedo
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider the static Maxwell system with an axially symmetric dielectric permittivity and construct complete systems of its solutions which can be used for analytic and numerical solution of corresponding boundary value problems.
    Mathematical Methods in the Applied Sciences 04/2007; · 0.78 Impact Factor
  • Source
    Sergei M. Grudsky, Kira V. Khmelnytskaya, Vladislav V. Kravchenko
    [Show abstract] [Hide abstract]
    ABSTRACT: Maxwell's equations for the time-dependent electromagnetic field in a homogeneous chiral medium are reduced to a single quaternionic equation. Its fundamental solution satisfying the causality principle is obtained which allows us to solve the time-dependent chiral Maxwell system with sources.
    Journal of Physics A General Physics 09/2003;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: V. D. Kupradse and M. A. Alexidse introduced the sixties of the last century a multipole-method based on fundamental solutions of differential operators, which was later applied by W. Freeden in geophysics. K. Gürlebeck and W. Sprößig found [K. Gürlebeck, W. Sprößig and M. Tasche, “Numerical realization of boundary collocation methods”, Multivariate approximation theory III, Proc. Conf., Oberwolfach/Ger. 1985, ISNM 75, 206–217 (1985; Zbl 0604.65088)] a corresponding collocation method in quaternion Sobolev spaces for a wide range of equations of mathematical physics. In the present paper the authors consider multipole methods for equations of Helmholtz type. For this reason the generalization to complex quaternions is necessary. A very interesting connection between quaternionic differential operators and Maxwell’s system for chiral media is deduced. Using this relaion the completeness in Sobolev spaces with suitable multipliers is shown for Maxwell’s equations in bounded and unbounded domains. In outer domains in case of the equation of Helmholtz type the well-known Sommerfeld condition is generalized and for Maxwell’s equations the electric and magnetic fields the Silver-Müller radiation condition is derived. The reader can find numerical examples at the end of the paper.
    Zeitschrift Fur Analysis Und Ihre Anwendungen - Z ANAL ANWEND. 01/2003;

Publication Stats

73 Citations
19.56 Total Impact Points

Institutions

  • 2010–2013
    • Autonomous University of Queretaro
      Ciudad Queretaro, Querétaro, Mexico
  • 2007–2010
    • Instituto Potosino de Investigación Científica y Tecnológica
      San Luis, San Luis Potosí, Mexico
  • 2003–2009
    • National Polytechnic Institute
      • Escuela Superior de Física y Matemáticas
      Mexico City, The Federal District, Mexico