Publications (27)23.5 Total impact
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ABSTRACT: The timedependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the onedimensional case reduces to a Vekuatype equation for bicomplexvalued functions of a hyperbolic variable (see arXiv:1001.0552). Using this relation we solve the problem of the transmission through an inhomogeneous layer of a normally incident electromagnetic timedependent plane wave. The solution is written in terms of a pair of Darbouxassociated transmutation operators (see arXiv:1111.4449), and combined with the recent results on their construction (see arXiv:1208.6166, arXiv:1306.2914) can be used for efficient computation of the transmitted modulated signals. We develop the corresponding numerical method and illustrate its performance with examples.  [Show abstract] [Hide abstract]
ABSTRACT: Let (a,b) be a finite interval and 1/p, q, r be functions from L1(a,b). We show that a general solution (in the weak sense) of the equation (pu')'+qu = zru on (a,b) can be constructed in terms of power series of the spectral parameter z. The series converge uniformly on [a,b] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed.Mathematical Methods in the Applied Sciences 09/2014; DOI:10.1002/mma.3282 · 0.88 Impact Factor  [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 SturmLiouville equations with variable coefficients. We consider SturmLiouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and FloquetBloch solutions; quantummechanical spectral and transmission problems as well as the eigenvalue problems for the ZakharovShabat 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 06/2014; 38(10). DOI:10.1002/mma.3213 · 0.88 Impact Factor  [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 nonshifted supersymmetric partners. The zeromode 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.  Mathematical Methods in the Applied Sciences 09/2013; 36(14):18781891. DOI:10.1002/mma.2732 · 0.88 Impact Factor
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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 KleinGordon 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 KleinGordon 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):191212. DOI:10.1016/j.jmaa.2012.10.013 · 1.12 Impact Factor  [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 nonshifted supersymmetric partners. The zeromode 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. DOI:10.1142/S0217732313400178 · 1.34 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A general solution of the fourthorder 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 fourthorder Sturm–Liouville equations. Several examples of application are discussed.Applied Mathematics and Computation 12/2012; 219(8):3610–3624. DOI:10.1016/j.amc.2012.09.055 · 1.60 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We determine the kind of parametric oscillators that are generated in the usual factorization procedure of secondorder 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 secondorder differential equations and the values of the shift parameter providing strictly periodic and antiperiodic solutions. We also notice that this simple problem presents paritytime (PT) symmetry. Possible applications are mentioned.Physics Letters A 09/2011; 375(40):34913495. DOI:10.1016/j.physleta.2011.08.024 · 1.63 Impact Factor  [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), 732734] 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 zeromode solutions. The fermionic equations can be considered as representing cosmologies of Stephani type, i.e., inhomogeneous and curvaturechanging in the conformal time. We next apply the socalled 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 paritytime (PT) property of the partner curvature indices.Symmetry Integrability and Geometry Methods and Applications 12/2010; DOI:10.3842/SIGMA.2011.013 · 1.30 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We determine the kind of parametric oscillators that are generated in the usual factorization procedure of secondorder 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 secondorder differential equations and the values of the shift parameter providing strictly periodic and antiperiodic solutions. We also notice that this simple problem presents paritytime (PT) symmetry. Possible applications are mentioned  [Show abstract] [Hide abstract]
ABSTRACT: We provide the representation of quasiperiodic 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 quasiexactly solvable periodic potentials  [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; DOI:10.1016/j.aop.2010.06.009 · 3.07 Impact Factor 
Article: Periodic SturmLiouville problems related to two Riccati equations of constant coefficients
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ABSTRACT: We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetricallycoupled secondorder 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 wellknown selfmatching 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. PhysAnnals of Physics 10/2009; 325(3). DOI:10.1016/j.aop.2009.12.002 · 3.07 Impact Factor  [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. DOI:10.1088/14644258/11/6/065707 · 1.92 Impact Factor  [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  [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 ErmakovLewis approach for the same model. This leads us to numerical calculations of the LewisRiesenfeld phases that could be of forthcoming experimental interest for bilayer graphene. We also present a generalization of the IoffeKorsch nonlinear Darboux transformation. PACS numbers: 02.30.Hq, 11.30.Pb, 81.05.Uw 1Journal of Physics A Mathematical and Theoretical 01/2009; 42(4). DOI:10.1088/17518113/42/4/042004 · 1.69 Impact Factor  [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 01/2009; 33(4). DOI:10.1002/mma.1210 · 0.88 Impact Factor  [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 01/2009; 33(4):469  472. DOI:10.1002/mma.1218 · 0.88 Impact Factor 
Conference Paper: Solution of the static Maxwell system for inhomogeneous media using generalized analytic function theory
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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
Publication Stats
132  Citations  
23.50  Total Impact Points  
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Institutions

2010–2014

Autonomous University of Queretaro
Ciudad Queretaro, Querétaro, Mexico


2009–2010

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
