American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An extension of the landweber regularization for a backward time fractional wave problem
  • DCDS-S Home
  • This Issue
  • Next Article
    Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria
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  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, 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 $ (- - -)
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[2]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353
[3]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175
[4]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[5]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
[6]

Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167
[7]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[8]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012
[9]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[10]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383
[11]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[12]

Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020388
[13]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093
[14]

Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376
[15]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[16]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[17]

Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1
[18]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003
[19]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[20]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (23)
  • HTML views (73)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences