Browsing by Author "Medina Reus, Elena"
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PublicationA classification of integrable quasiclassical deformations of algebraic curves(IOP Publishing, 2006-08-08) Konopelchenko, Boris; Martínez Alonso, Luis; Medina Reus, ElenaA previously introduced scheme for describing integrable deformations of algebraic curves is completed. Lenard relations are used to characterize and classify these deformations in terms of hydrodynamic-type systems. A general solution of the compatibility conditions for consistent deformations is given and expressions for the solutions of the corresponding Lenard relations are provided. PublicationA Common integrable structure in the hermitian matrix model and Hele-Shaw flows(Atlantis Press, 2008-10) Martínez Alonso, Luis; Medina Reus, ElenaIt is proved that the system of string equations of the dispersionless 2-Toda hierarchy which arises in the planar limit of the hermitian matrix model also underlies certain processes in HeleShaw flows. PublicationA mathematical model for the growth of elongated bones(Italian Society for Applied and Industrial Mathematics (SIMAI), 2010) Fasano, A.; Herrero, Miguel A.; López, J. M.; Medina Reus, ElenaA mathematical model to describe the process of formation of bone tissue by replacement of cartilage tissue is presented and discussed. This model is based on an absorption-diffusion system which describes the interaction of two key signalling molecules. These molecules characterize the dynamics of the transition zone between the cartilage and the bone tissue. Some experimental data are needed to estimate some model parameters. We discuss how our results are essentially unaffected by small variations, and in a particular case, not necessarily small variations in the experimental values. PublicationAn efficient method for computing genus expansions and counting numbers in the Hermitian matrix model(Elsevier Science BV, 2011-07-11) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, ElenaWe present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher-Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach. PublicationAnalysis of the large-n limit of multiple orthogonal polynomials and non-intersecting Brownian motions(Amer Inst Physics, 2009) Martínez Alonso, Luis; Medina Reus, ElenaIn this work we use Riemann-Hilbert problems for multiple orthogonal polynomials in order to derive string equations associated to Lax-Orlov pairs operators. These string equations provide us with a useful tool to analyze the large n-limit of the related hierarchies. The results are finally applied to the study of the associated random matrix models (Gaussian Hermitian matrix models with an external source) and non-intersecting Brownian motions starting from a fix point. PublicationDetermination of S-curves with applications to the theory of nonhermitian orthogonal polynomials(2013-06) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, ElenaThis paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model. PublicationExact solutions of integrable 2D contour dynamics(Elsevier Science BV, 2005-02-24) Martínez Alonso, Luis; Medina Reus, ElenaA class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation is obtained. These solutions represent deformations of analytic curves with a finite number of nonzero harmonic moments. The corresponding tau-functions are determined and the emergence of cusps is studied. PublicationExplicit solutions of supersymmetric KP hierarchies: Supersolitons and solitinos(American Institute of Physics, 1996-12) Ibort, A.; Martínez Alonso, Luis; Medina Reus, ElenaWide classes of explicit solutions of the Manin-Radul and Jacobian supersymmetric KP hierarchies are constructed by using line bundles over complex supercurves based on the Riemann sphere. Their construction extends several ideas of the standard KP theory, such as wave functions,δ̅ equations and τ-functions. Thus, supersymmetric generalizations of N-soliton solutions, including a new purely odd ‘‘solitino’’ solution, as well as rational solutions, are found and characterized. PublicationFine structure in the large n limit of the non-hermitian Penner matrix model(Academic Press Inc Elsevier Science, 2015-10) Álvarez, Gabriel; Martínez Alonso, Luis; Medina Reus, ElenaIn this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large n limit in the non-hermitian Penner matrix model. In these generalizations g_(n)n → t, but the product g_(n)n is not necessarily fixed to the value of the ’t Hooft coupling t. If t > 1 and the limit l = lim_(n→∞) |sin(π/g_n)| ^(1/n) exists, then the large n limit is well-defined but depends both on t and on l. This result implies that for t > 1 the standard large n limit with g_(n)n = t fixed is not well-defined. The parameter l determines a fine structure of the asymptotic eigenvalue support: for l ≠ 0 the support consists of an interval on the real axis with charge fraction Q = 1 − 1/t and an l-dependent oval around the origin with charge fraction 1/t. For l = 1 these two components meet, and for l = 0 the oval collapses to the origin. We also calculate the total electrostatic energy Ԑ which turns out to be independent of l, and the free energy Ƒ = Ԑ - Ǫ ln l, which does depend of the fine structure parameter l. The existence of large n asymptotic expansions of Ƒ beyond the planar limit as well as the double-scaling limit are also discussed. PublicationFinite-time aggregation into a single point in a reaction-diffusion system(IOP Publishing Ltd, 1997-11) Herrero, Miguel A.; Medina Reus, Elena; Velázquez, J.J. L.We consider the following system: [GRAPHICS] which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dine mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed. PublicationFormalismo fermiónico de solitones en modelos integrables de la dinámica de los fluidos(Universidad Complutense de Madrid, Servicio de Publicaciones, 2002) Medina Reus, Elena; Martínez Alonso, Luis PublicationGenus-zero Whitham hierarchies in conformal-map dynamics(Elsevier Science BV, 2006-08-26) Martínez Alonso, Luis; Medina Reus, ElenaA scheme for solving quasiclassical-string equations is developed to prove that genus-zero hitham hierarchies describe the deformations of planar domains determined by rational conformal-maps. This property is applied in normal matrix models to show that deformations of simply-connected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies. PublicationGravitational lensing by eigenvalue distributions of random matrix models(IOP Publishing Ltd, 2018-05-10) Martínez Alonso, Luis; Medina Reus, ElenaWe propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles. PublicationIntegrable quasiclassical deformations of cubic curves(American Institute of Physics, 2005-11) Kodama, Y.; Konopelchenko, Boris; Martínez Alonso, Luis; Medina Reus, ElenaA general scheme for determining and studying hydrodynamic type systems describing integrable deformations of algebraic curves is applied to cubic curves. Lagrange resolvents of the theory of cubic equations are used to derive and characterize these deformations. PublicationKinetic dominance and psi series in the Hamilton-Jacobi formulation of inflaton models(Amer Physical Soc, 2020-11-13) Medina Reus, Elena; Martínez Alonso, LuisSingle-field inflaton models in the kinetic dominance period admit formal solutions given by generalized asymptotic expansions called psi series. We present a method for computing psi series for the Hubble parameter as a function of the inflaton field in the Hamilton-Jacobi formulation of inflaton models. Similar psi series for the scale factor, the conformal time, and the Hubble radius are also derived. They are applied to determine the value of the inflaton field when the inflation period starts and to estimate the contribution of the kinetic dominance period to calculate the duration of inflation. These psi series are also used to obtain explicit two-term truncated psi series near the singularity for the potentials of the Mukhanov-Sasaki equation for curvature and tensor perturbations. The method is illustrated with wide families of inflaton models determined by potential functions combining polynomial and exponential functions, as well as with generalized Starobinsky models. PublicationKinetic dominance and the wave function of the Universe(Amer Physical Soc, 2022-04-06) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, ElenaWe analyze the emergence of classical inflationary universes in a kinetic-dominated stage using a suitable class of solutions of the Wheeler-DeWitt equation with a constant potential. These solutions are eigenfunctions of the inflaton momentum operator that are strongly peaked on classical solutions exhibiting either or both a kinetic-dominated period and an inflation period. Our analysis is based on semiclassical WKB solutions of the Wheeler-DeWitt equation interpreted in the sense of Borel (to perform a correct connection between classically allowed regions) and on the relationship of these solutions to the solutions of the classical model. For large values of the scale factor the WKB Vilenkin tunneling wave function and the Hartle-Hawking no-boundary wave functions are recovered as particular instances of our class of wave functions. PublicationLarge N expansions and Painlevé hierarchies in the Hermitian matrix model(IOP Publishing Ltd, 2011-07-15) Álvarez Galindo, Gabriel; Martínez Alonso, Luis; Medina Reus, ElenaWe present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlevé I and Painlevé II hierarchies associated with one-cut and two-cut critical models, respectively. PublicationLocalized coherent structures of the Davey-Stewartson equation in the bilinear formalism(American Institute of Physics, 1992-09) Martínez Alonso, Luis; Medina Reus, ElenaThe DaveyStewartson equation is considered from the point of view of the bilinear formalism of the Kyoto school. Multidromion solutions are constructed in terms of free fermions and their asymptotic properties are characterized. The dynamical properties of dromions are discussed. PublicationMultiple orthogonal polynomials, string equations and the large-n limit(IOP Publishing Ltd, 2009-03-22) Martínez Alonso, Luis; Medina Reus, ElenaThe Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated with pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the phase space of the Whitham hierarchy of dispersionless integrable systems is provided. Applications to the analysis of the large-n limit of multiple orthogonal polynomials and their associated random matrix ensembles and models of non-intersecting Brownian motions are given. PublicationOn the dynamics of the growth plate in primary ossification(Elsevier, 2010-08-21) Fasano, A.; Herrero, Miguel A.; López, J. M.; Medina Reus, ElenaIn this work a mathematical model for the interaction of two key signalling molecules in rat tibia ossification is presented and discussed. The molecules under consideration are Indian hedgehog (Ihh) and parathyroid hormone-related peptide (PTHrP). These are known to be major agents in the dynamics of the so-called growth plate, where transition from pristine cartilage to advancing bone takes place. Our model consists in a steady-state linear approximation to a reaction-diffusion system where only diffusion and absorption mechanisms are retained. Estimates on some system parameters are given, on the basis of the knowledge of a few measurable quantities. This allows for explicitly solving our model, whereupon a discussion on robustness and regulatory properties thereof is provided. In particular, we show that the size of the Proliferative Zone in the growth plate is rather insensitive to variations in the flux coefficients for Ihh and PTHrP at their boundaries. Besides, we also show that the model is also insensitive to large changes in the (comparatively small) critical value of the PTHrP concentration which marks the transition form Proliferative to Hyperthropic Regions within the Growth Plate. These results hold irrespective of the particular diffusivities selected for Ihh and PTHrP.