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
    The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data
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 $ (- - -)
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]

Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021204
[2]

Guanggan Chen, Qin Li, Yunyun Wei. Approximate dynamics of a class of stochastic wave equations with white noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021033
[3]

Angelo Favini, Georgy A. Sviridyuk, Alyona A. Zamyshlyaeva. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure & Applied Analysis, 2016, 15 (1) : 185-196. doi: 10.3934/cpaa.2016.15.185
[4]

Johannes Eilinghoff, Roland Schnaubelt. Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5685-5709. doi: 10.3934/dcds.2018248
[5]

Xinfu Chen, Carey Caginalp, Jianghao Hao, Yajing Zhang. Effects of white noise in multistable dynamics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1805-1825. doi: 10.3934/dcdsb.2013.18.1805
[6]

Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
[7]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[8]

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
[9]

Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032
[10]

Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056
[11]

Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021133
[12]

Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381
[13]

Boris P. Belinskiy, Peter Caithamer. Stochastic stability of some mechanical systems with a multiplicative white noise. Conference Publications, 2003, 2003 (Special) : 91-99. doi: 10.3934/proc.2003.2003.91
[14]

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
[15]

Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873
[16]

Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941
[17]

Boris P. Belinskiy, Peter Caithamer. Energy of an elastic mechanical system driven by Gaussian noise white in time. Conference Publications, 2001, 2001 (Special) : 39-49. doi: 10.3934/proc.2001.2001.39
[18]

Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks & Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008
[19]

Santiago Cano-Casanova. Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations. Conference Publications, 2011, 2011 (Special) : 240-249. doi: 10.3934/proc.2011.2011.240
[20]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (127)
  • HTML views (289)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences