Figure 1.
Graphical representation of the function $ f $ (-) and the convex upper bound $ g $ (- - -)
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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
| 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 |
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) $ |
| $ L^1_x\cap H^1_x $ |
| $ L^\infty_x $ |
| $ O(t^{-\frac14}) $ |
| $ t $ |
| $ +\infty $ |
| $ p>7 $ |
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 and 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.
|
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
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.
|
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
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
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