2024 Volume 14 Issue 2
Article Contents
Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(2): 886-910
Next Article Previous Article

Daniele C. R. Gomes, Mauro A. Rincon, Maria Darci G. da Silva, Gladson O. Antunes. NUMERICAL AND MATHEMATICAL ANALYSIS OF A NONLINEAR SCHRÖDINGER PROBLEM WITH MOVING ENDS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 886-910. doi: 10.11948/20230189
Citation: Daniele C. R. Gomes, Mauro A. Rincon, Maria Darci G. da Silva, Gladson O. Antunes. NUMERICAL AND MATHEMATICAL ANALYSIS OF A NONLINEAR SCHRÖDINGER PROBLEM WITH MOVING ENDS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 886-910. doi: 10.11948/20230189
shu

NUMERICAL AND MATHEMATICAL ANALYSIS OF A NONLINEAR SCHRÖDINGER PROBLEM WITH MOVING ENDS

  • 1.

    Instituto de Computação, Universidade Federal do Rio de Janeiro, RJ, Brazil

  • 2.

    Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil

  • 3.

    Escola de Matemática, Universidade Federal do Estado do Rio de Janeiro, RJ, Brazil

    Abstract

  • Existence and uniqueness of solution in $ \mathbb{R}^n $ for a nonlinear Shrödinger equation in a domain with moving ends and an optimal algorithm to obtain an approximate numerical solution for the two-dimensional space. The linearized Crank-Nicolson-Galerkin method is proposed to achieve high-performance computing. Numerical examples for the two-dimensional domain are presented to confirm the theoretical analysis and numerical results. Numerical errors associated with the linear, quadratic and cubic base are displayed.

    MSC: 35K55, 65M06, 65M60, 35R37
    • FullText(HTML)
  • 加载中
    • References
  • [1] R. M. P. Almeida, J. C. M. Duque, J. Ferreira and R. J. Robalo, Finite element schemes for a class of nonlocal parabolic systems with moving boundaries, Journal of Applied Numerical Mathematics, 2018, 127, 226–248. DOI: 10.1016/j.apnum.2018.01.007.

    CrossRef Google Scholar

    [2] G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM Journal on Numerical Analysis, 1976, 13(4), 564–576. DOI: 10.2307/2156246.

    CrossRef Google Scholar

    [3] J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation, Journal of Mathematical Physics, 1984, 25, 3270–3273. DOI: 10.1063/1.526074.

    CrossRef Google Scholar

    [4] C. Besse, A relaxation scheme for the nonlinear Schrodinger equation, SIAM Journal on Numerical Analysis, 2004, 42(3), 934–952. DOI: 10.1137/S0036142901396521.

    CrossRef Google Scholar

    [5] P. Cannarsa, G. da Prato, and J. P. Zolesio, The damped wave equation in a moving domain, Journal of Differential Equations, 1990, 85(1), 1–16. doi: 10.1016/0022-0396(90)90086-5

    CrossRef Google Scholar

    [6] X. Carvajal, M. Panthee and M. Scialom, On well-posedness of the third-order nonlinear Schrodinger equation with time-dependent coefficients, Communications in Contemporary Mathematics, 2015, 17(04). DOI: 10.1142/S021919971450031X.

    CrossRef Google Scholar

    [7] T. Cazenave, Semilinear Schrodinger Equations, Courant Lecture Notes in Mathematics, AMS, New York, 2003, 10.

    Google Scholar

    [8] F. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in applied mathematics, SIAM, Paris, France, 2002, 40.

    Google Scholar

    [9] R. Cipolatti, J. L. Gondar and C. Trallero-Giner, Bose-Einstein condensates in optical lattices: Mathematical analysis and analytical approximate formulae, Physica D: Nonlinear Phenomena, Amsterdam, 2012, 241(7), 755–763. DOI: 10.1016/j.physd.2011.12.012.

    CrossRef Google Scholar

    [10] H. R. Clark, M. A. Rincon and R. D. Rodrigues, Beam equation with weak-internal damping in domain with moving boundary, Journal of Applied Numerical Mathematics, 2003, 47(2), 139–157. doi: 10.1016/S0168-9274(03)00066-7

    CrossRef Google Scholar

    [11] F. Dalfovo and S. Giorgini, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, 1999, 71(3), 463–512. doi: 10.1103/RevModPhys.71.463

    CrossRef Google Scholar

    [12] D. C. R. Gomes, M. A. Rincon, M. D. G. Silva and G. O. Antunes, Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary, Advances in Computational Mathematics, Springer US, 2019, 45(4), 981–1004. DOI: 10.1007/s10444-018-9643-3.

    Google Scholar

    [13] D. F. Griffiths, R. Mitchell and J. L. Morris, A numerical study of the nonlinear Schrödinger equation, Computer Methods in Applied Mechanics and Engineering, Springer US, 1984, 45, 177–205. DOI: 10.1016/0045-7825(84)90156-7.

    Google Scholar

    [14] D. Hartree, The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods, Proc. Camb. Philos. Soc., 1968, (24), 89–132.

    Google Scholar

    [15] P. Henning and J. Wärnegärd, A note on optimal $ H^1 $-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation, BIT Numerical Mathematics, 2020. DOI: 10.1007/s10543-020-00814-3.

    CrossRef $ H^1 $-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation" target="_blank">Google Scholar

    [16] L. I. Ignat and E. Zuazua, Convergence rates for dispersive approximation schemes to nonlinear Schrodinger equations, Journal de Mathématiques Pures et Appliquées, 2012, 98(5), 479–517. doi: 10.1016/j.matpur.2012.01.001

    CrossRef Google Scholar

    [17] J. Jina and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrodinger equation on unbounded domain, Journal of Computational and Applied Mathematics, 2008, 220, 240–256. doi: 10.1016/j.cam.2007.08.006

    CrossRef Google Scholar

    [18] P. L. Kelley, Self-focusing of optical beams, Phys. Rev. Lett., 1965, 15(26), 1005–1008. DOI: 10.1103/PhysRevLett.15.1005.

    CrossRef Google Scholar

    [19] J. L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites non Linéaires, Dunod, Paris, 1969.

    Google Scholar

    [20] I. -S. Liu and M. A. Rincon, Effect of moving boundaries on the vibrating elastic string, Applied Numerical Mathematics, 2003, 47(2), 159–172. doi: 10.1016/S0168-9274(03)00063-1

    CrossRef Google Scholar

    [21] L. A. Medeiros, J. Límaco and S. B. Menezes, Vibrations of elastic strings: mathematical aspects, Journal of Computational Analysis and applications, 2002, 4, 211–263. doi: 10.1023/A:1013151525487

    CrossRef Google Scholar

    [22] Y. Mei and R. An, Error estimates of second-order BDF Galerkin finite element methods for a coupled nonlinear Schrödinger system, Computers & Mathematics with Applications, 2022, 122, 117–125. DOI: 10.1016/j.camwa.2022.07.018.

    CrossRef Google Scholar

    [23] M. A. Rincon and M. O. Lacerda, Error analysis of thermal equation with moving ends, Journal of Mathematics and Computers in Simulation, 2010, 80(11), 2200–2215. doi: 10.1016/j.matcom.2010.04.013

    CrossRef Google Scholar

    [24] M. A. Rincon, I. -S. Liu, W. R. Huarcaya and B. A. Carmo, Numerical analysis for a nonlinear model of elastic strings with moving ends, Applied Numerical Mathematics, 2019, 135, 146–164. DOI: 10.1016/j.apnum.2018.08.014.

    CrossRef Google Scholar

    [25] M. Sepúlveda and O. Vera, Numerical methods for a coupled nonlinear Schrodinger system, Bol. Soc. Esp. Mat. Apl., 2008, 43, 97–104.

    Google Scholar

    [26] W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Mathematical Physics, 1974, 53–78. DOI: 10.1007/978-94-010-2147-0_3.

    CrossRef Google Scholar

    [27] K. E. Strecker, G. B. Partridge, A. G. Truscott and R. G. Hulet, Bright matter wave solitons in Bose-Einstein condensates, New Journal of Physics, 2003, 5(73), 1–8.

    Google Scholar

    [28] V. Thomée, Galerkin finite element methods for parabolic problems, Springer, Springer Series in Computational Mathematics, Rio de Janeiro, 2006, 25(2).

    Google Scholar

    [29] D. A. Vigo, J. C. Xavier and M. A. Rincon, Analysis and computation of a nonlinear Korteweg-de Vries system, BIT Numerical Mathematics, 2016, 56(3), 1069–1099. DOI: 10.1007/s10543-015-0589-2.

    CrossRef Google Scholar

    [30] J. Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrodinger equation, Journal of Scientific Computing, 2014, 60, 390–407. DOI: 10.1007/s10915-013-9799-4.

    CrossRef Google Scholar

    [31] T. Wang, Uniform point-wise error estimates of semi-implicit compact finite difference methods for the nonlinear Schrodinger equation perturbed by wave operator, Journal of Computational and Applied Mathematics, 2015, 422(1), 286–308.

    Google Scholar

    [32] M. L. Wheeler, A priori $ L_2 $ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM Journal on Numerical Analysis, 1973, 10(4), 723–759. doi: 10.1137/0710062

    CrossRef $ L_2 $ error estimates for Galerkin approximations to parabolic partial differential equations" target="_blank">Google Scholar

    • Supplements
    • Cited by

Figures(8)

Export Citation
PDF
XML

Article Metrics

Article views(1267) PDF downloads(306) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Published by Wilmington Scientific Publisher