Advanced Search
Article Contents
Article Contents

On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we establish a convergence result for the operator splitting scheme $ Z_{\tau} $ introduced by Ignat [12], with initial data in $ H^1 $, for the nonlinear Schrödinger equation:

    $ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) = \phi (x), $

    where $ p >0 $, $ \lambda \in \{-1,1\} $ and $ (x,t) \in \mathbb{R}^d \times [0,\infty) $. We prove the $ L^2 $ convergence of order $ \mathcal{O}(\tau^{1/2}) $ for the scheme with initial data in the space $ H^1 (\mathbb{R}^d) $ for the energy-subcritical range of $ p $.

    Mathematics Subject Classification: Primary:35Q55, 65M15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] R. Altmann and A. Ostermann, Splitting methods for constrained diffusion-reaction systems, Comput. Math. Appl., 74 (2017), 962-976.  doi: 10.1016/j.camwa.2017.02.044.
    [2] C. BesseB. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 26-40.  doi: 10.1137/S0036142900381497.
    [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, Amer. Math. Soc/Courant Institute of Mathematical Sciences, Providence, RI/York New, NY, 2003. doi: 10.1090/cln/010.
    [4] M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.
    [5] J. EilinghoffR. Schnaubelt and K. Schratz, Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation, J. Math. Anal. Appl., 442 (2016), 740-760.  doi: 10.1016/j.jmaa.2016.05.014.
    [6] E. FaouA. Ostermann and K. Schratz, Analysis of exponential splitting methods for inhomogeneous parabolic equations, IMA J. Numer. Anal., 35 (2015), 161-178.  doi: 10.1093/imanum/dru002.
    [7] L. Gauckler and C. Lubich, Splitting integrators for nonlinear Schrödinger equations over long times, Found. Comput. Math., 10 (2010), 275-302.  doi: 10.1007/s10208-010-9063-3.
    [8] L. Gauckler and C. Lubich, Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169.  doi: 10.1007/s10208-010-9059-z.
    [9] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle River, NJ, 2004.
    [10] E. Hansen and A. Ostermann, High-order splitting schemes for semilinear evolution equations (English summary), Bit Numer. Math., 56 (2016), 1303-1316.  doi: 10.1007/s10543-016-0604-2.
    [11] L. I. Ignat, Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591.  doi: 10.1142/S0218202507002029.
    [12] L. I. Ignat, A splitting method for the nonlinear Schrödinger equation, J. Differential Equations, 250 (2011), 3022-3046.  doi: 10.1016/j.jde.2011.01.028.
    [13] L. I. Ignat and E. Zuazua, A two-grid approximation scheme for nonlinear Schrödinger equations: Dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386.  doi: 10.1016/j.crma.2005.07.018.
    [14] L. I. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390.  doi: 10.1137/070683787.
    [15] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.
    [16] M. KnöllerA. Ostermann and K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data (English summary), SIAM J. Numer. Anal., 57 (2019), 1967-1986.  doi: 10.1137/18M1198375.
    [17] J. Lu and J. L. Marzuola, Strang splitting methods for a quasilinear Schrödinger equation: Convergence, instability, and dynamics, Commun. Math. Sci., 13 (2015), 1051-1074.  doi: 10.4310/CMS.2015.v13.n5.a1.
    [18] C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equation, Math. Comp., 77 (2008), 2141-2153.  doi: 10.1090/S0025-5718-08-02101-7.
    [19] A. Ostermann, F. Rousset and K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Found. Comput. Math., (2020), to appear.
    [20] A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.  doi: 10.1007/s10208-017-9352-1.
    [21] K. SchratzY. Wang and X. Zhao, Low-regularity integrators for nonlinear Dirac equations, Math. Comp., 90 (2021), 189-214.  doi: 10.1090/mcom/3557.
    [22] R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.
    [23] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.
    [24] T. Tao, Nonlinear Dispersive Equations. Local And Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.
    [25] M. Thalhammer, Higher-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), 2022-2038.  doi: 10.1137/060674636.
    [26] M. ThalhammerM. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.  doi: 10.1016/j.jcp.2008.10.008.
  • 加载中

Article Metrics

HTML views(423) PDF downloads(296) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint