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August  2021, 14(8): 2877-2891. doi: 10.3934/dcdss.2020456

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

Serge Dumont 1,, , Olivier Goubet 2, and  Youcef Mammeri 3,

1. 

IMAG UMR 5149 CNRS, Université de Nîmes, Place Gabriel Péri, 30000 Nîmes, France
2. 

Laboratoire Paul Painlevé CNRS UMR 8524, et équipe projet INRIA PARADYSE, Université de Lille, 59 655 Villeneuve d'Ascq cedex, France
3. 

LAMFA UMR 7352 CNRS, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039 Amiens, France

* Corresponding author: serge.dumont@unimes.fr

This article is dedicated to the memory of Ezzeddine Zahrouni.

Received  February 2020 Revised  September 2020 Published  August 2021 Early access  November 2020

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 $
.
Keywords: NLS equations, white noise dispersion, decay rate, splitting scheme.
Mathematics Subject Classification: Primary:35Q55;Secondary:65C30, 60H35.
Citation: Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2877-2891. doi: 10.3934/dcdss.2020456
References:
[1]

P. Antonelli, J.-C. Saut and C. Sparber, Well-posedness and averaging of NLS with time-periodic dispersion management, Adv. Diff. Eq., 18 (2013), 49-68.   Google Scholar

[2]

R. Belaouar, A. 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.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[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.  Google Scholar

[5]

M. Chen, O. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

G. Fenger, O. 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.  Google Scholar

[12]

N. Hayashi, E. Kaikina, P. Naumkin and A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884. Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

P. Antonelli, J.-C. Saut and C. Sparber, Well-posedness and averaging of NLS with time-periodic dispersion management, Adv. Diff. Eq., 18 (2013), 49-68.   Google Scholar

[2]

R. Belaouar, A. 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.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[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.  Google Scholar

[5]

M. Chen, O. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

G. Fenger, O. 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.  Google Scholar

[12]

N. Hayashi, E. Kaikina, P. Naumkin and A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884. Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Graphical representation of the function $ f $ (-) and the convex upper bound $ g $ (- - -)
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Figure 2.  $ L^2 $ convergence with respect to the time step of discretization $ \Delta t $
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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)
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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)
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Figure 5.  $ L^\infty $ decay rate with respect to time for the deterministic and the stochastic problem
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