\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion

  • * Corresponding author: serge.dumont@unimes.fr

    * Corresponding author: serge.dumont@unimes.fr 

This article is dedicated to the memory of Ezzeddine Zahrouni.

Abstract Full Text(HTML) Figure(5) Related Papers Cited by
  • In this article, the asymptotic behavior of the solution to the following one dimensional Schrödinger equations with white noise dispersion

    $ idu + u_{xx}\circ dW+ |u|^{p-1}udt = 0 $

    is studied. Here the equation is written in the { Stratonovich} formulation, and $ W(t) $ is a standard real valued Brownian motion. After establishing the global well-posedness, theoretical proof and numerical investigations are provided showing that, for a deterministic small enough initial data in $ L^1_x\cap H^1_x $, the expectation of the $ L^\infty_x $ norm of the solutions decay to zero at $ O(t^{-\frac14}) $ as $ t $ goes to $ +\infty $, as soon as $ p>7 $.

    Mathematics Subject Classification: Primary:35Q55;Secondary:65C30, 60H35.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Graphical representation of the function $ f $ (-) and the convex upper bound $ g $ (- - -)

    Figure 2.  $ L^2 $ convergence with respect to the time step of discretization $ \Delta t $

    Figure 3.  Space and time evolution of the approximate solution of the nonlinear equation with $ p = 5 $ for one stochastic process (left: real part; right: imaginary part)

    Figure 4.  Space and time evolution of the approximate solution of the nonlinear equation with $ p = 13 $ for one stochastic process (left: real part; right: imaginary part)

    Figure 5.  $ L^\infty $ decay rate with respect to time for the deterministic and the stochastic problem

  • [1] P. AntonelliJ.-C. Saut and C. Sparber, Well-posedness and averaging of NLS with time-periodic dispersion management, Adv. Diff. Eq., 18 (2013), 49-68. 
    [2] R. BelaouarA. de Bouard and A. Debussche, Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion, A. Stoch PDE: Anal Comp, 3 (2015), 103-132.  doi: 10.1007/s40072-015-0044-z.
    [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.
    [4] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
    [5] M. ChenO. Goubet and Y. Mammeri, Generalized regularized long waves equations with white noise dispersion, Stoch. Partial Differ. Equ. Anal. Comput., 5 (2017), 319-342.  doi: 10.1007/s40072-016-0089-7.
    [6] K. Chouk and M. Gubinelli, Nonlinear PDEs with modulated dispersion â… : Nonlinear Schrödinger equations, Comm. Partial Differential Equations, 40 (2015), 2047-2081.  doi: 10.1080/03605302.2015.1073300.
    [7] A. de Bouard and A. Debussche, The nonlinear Schrödinger equation with white noise dispersion, J. Func. Anal., 259 (2010), 1300-1321.  doi: 10.1016/j.jfa.2010.04.002.
    [8] A. Debussche and Y. Tsutsumi, 1D quintic nonlinear Schrodinger equation with white noise dispersion, J. Math. Pures Appli., 96 (2011), 363-376.  doi: 10.1016/j.matpur.2011.02.002.
    [9] R. Duboscq and R. Marty, Analysis of a splitting scheme for a class of random nonlinear partial differential equations, ESAIM: PS, 20 (2016), 572-589.  doi: 10.1051/ps/2016023.
    [10] R. Duboscq and A. Reveillac, On a stochastic Hardy-Littlewood-Sobolev inequality with application to Strichartz estimates for a noisy dispersion, arXiv: 1711.07188v1 [math.AP], 2017.
    [11] G. FengerO. Goubet and Y. Mammeri, Numerical analysis of the midpoint scheme for the generalized Benjamin-Bona-Mahony equation with white noise dispersion, CiCP, 26 (2019), 1397-1414.  doi: 10.4208/cicp.2019.js60.02.
    [12] N. Hayashi, E. Kaikina, P. Naumkin and A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884. Springer-Verlag, Berlin, 2006.
    [13] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.
    [14] R. Marty, On a splitting scheme for the nonlinear Schrödinger equation in a random medium, Comm. Math. Sci., 4 (2006), 679-705.  doi: 10.4310/CMS.2006.v4.n4.a1.
  • 加载中

Figures(5)

SHARE

Article Metrics

HTML views(370) PDF downloads(261) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return