Examinando por Materia "ECUACIONES DIFERENCIALES NO LINEALES"

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  • Proyecto final de Grado
    Implementación numérica de un modelo de fracturación hidráulica mediante XFEM
    (2018) De Cunto, Luis Octavio; D’hers, Sebastián
    "La técnica de fracturación hidráulica es un proceso de importante relevancia en la extracción de petróleo y gas. Resumidamente, este proceso consiste en inyectar un fluido a muy alta presión en una roca para fracturarla y, de esta manera, aumentar su permeabilidad. Físicamente, este proceso involucra tres problemas diferentes: Las variaciones de presión existentes en el fluido dentro de la fractura, la deformación de la roca debido a dicha presión ejercida por el fluido, y la propagación de la fractura. Este trabajo tiene como objetivo modelar numéricamente dicho proceso, teniendo en cuenta los tres fenómenos nombrados previamente. Al depender mutuamente, las ecuaciones que describen dichos fenómenos se encuentran acopladas, formando un sistema de ecuaciones no Lineal. Es por esto que se resuelven las ecuaciones que describen cada fenómeno iterativamente, tratando de modelar cada parte del problema en forma independiente. Se utiliza el método XFEM para modelar adecuadamente la fractura, ya que no es necesario que la malla se encuentre alineada con la fractura y, por otra parte, evita el problema de remallar cuando la fractura avanza."
  • Artículo de Publicación Periódica
    Measuring self-steepening with the photon-conserving nonlinear Schrödinger equation
    (2020-08-15) Linale, N.; Fierens, Pablo Ignacio; Bonetti, Juan I.; Sánchez, Alfredo D.; Hernández, Santiago M.; Grosz, Diego
    "We propose an original, simple, and direct method to measure self-steepening (SS) in nonlinear waveguides. Our proposal is based on results derived from the recently introduced photon-conserving nonlinear Schrödinger equation (NLSE) and relies on the time shift experienced by soliton-like pulses due to SS upon propagation. In particular, a direct measurement of this time shift allows for a precise estimation of the SS parameter. Furthermore, we show that such an approach cannot be tackled by resorting to the NLSE. The proposed method is validated through numerical simulations, in excellent agreement with the analytical model, and results are presented for relevant spectral regions in the near infrared, the telecommunication band, and the mid infrared, and for realistic parameters of available laser sources and waveguides. Finally, we demonstrate the robustness of the proposed scheme against deviations expected in real-life experimental conditions, such as pulse shape, pulse peak power, pulsewidth, and/or higher-order linear and nonlinear dispersion."
  • Artículo de Publicación Periódica
    Modified nonlinear Schrödinger equation for frequency-dependent nonlinear profiles of arbitrary sign
    (2019) Bonetti, Juan I.; Linale, N.; Sánchez, Alfredo D.; Hernández, Santiago M.; Fierens, Pablo Ignacio; Grosz, Diego
    In recent times, materials exhibiting frequency-dependent optical nonlinearities, such as nanoparticle-doped glasses and other metamaterials, have gathered significant interest. The simulation of the propagation of intense light pulses in such media, by means of the nonlinear Schrödinger equation (NLSE), poses the problem in that straightforward inclusion of a frequency-dependent nonlinearity may lead to unphysical results, namely, neither the energy nor the photon number is conserved in general. Inspired by a simple quantum-mechanical argument, we derive an energy- and photon-conserving NLSE (pcNLSE). Unlike others, our approach relies only on the knowledge of the frequency-dependent nonlinearity profile and a generalization of Miller’s rule for nonlinear susceptibility, enabling the simulation of nonlinear profiles of arbitrary frequency dependence and sign. Moreover, the proposed pcNLSE can be efficiently solved by the same numerical techniques commonly used to deal with the NLSE. Relevant simulation results supporting our theoretical approach are presented.
  • Artículo de Publicación Periódica
    Soliton solutions and self-steepening in the photon-conserving nonlinear Schrödinger equation
    (2020-12-09) Hernández, Santiago M.; Bonetti, Juan I.; Linale, N.; Grosz, Diego; Fierens, Pablo Ignacio
    "We have recently introduced a new modeling equation for the propagation of pulses in optical waveguides, the photon-conserving Nonlinear Schrödinger Equation (pcNLSE) which, unlike the canonical NLSE, guarantees strict conservation of both the energy and the number of photons for any arbitrary frequency-dependent nonlinearity. In this paper, we analyze some properties of this new equation in the familiar case where the nonlinear coefficient of the waveguide does not change sign. We show that the pcNLSE effectively adds a correction term to the NLSE proportional to the deviation of the self-steepening (SS) parameter from the photon-conserving condition in the NLSE. Furthermore, we describe the role of the self-steepening parameter in the context of the conservation of the number of photons and derive an analytical expression for the relation of the SS parameter with the time delay experienced by pulses upon propagation. Finally, we put forth soliton-like solutions of the pcNLSE that, unlike NLSE solitons, conserve the number of photons for any arbitrary SS parameter. "
  • Artículo de Publicación Periódica
    A state estimation strategy for a nonlinear switched system with unknown switching signals
    (2021) Benítez, Oscar; García Galiñanes, Rafael
    "A strategy is presented to estimate the state of a nonlinear autonomous switched system, with no knowledge of the switching signal, except its dwell time. To do so, algorithms to estimate the switching times and the current mode of the system are developed. The estimation of the switching times is based on approximating the second ( generalised) derivative of the output of the system via a convolution of this signal with a suitable function and on detecting the corresponding spikes. To estimate the modes, a scheme based on the use of a bank of observers (one for each mode) and of a bank of subsystems (for each step of the estimation process a suitable subset of the subsystems of the switched system) is developed. The algorithms run regardless of the state observer model, as long as its output error norm decays exponentially with a controlled decay rate."
  • Artículo de Publicación Periódica
    Uniform stability properties of switched systems with switchings governed by digraphs
    (2005-05) Mancilla-Aguilar, J. L.; García Galiñanes, Rafael; Sontag, E.; Wang, Y.
    "Characterizations of various uniform stability properties of switched systems described by differential inclusions, and whose switchings are governed by a digraph, are developed. These characterizations are given in terms of stability properties of the system with restricted switchings and also in terms of Lyapunov functions."