
-
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
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 |
$ idu + u_{xx}\circ dW+ |u|^{p-1}udt = 0 $ |
$ W(t) $ |
$ L^1_x\cap H^1_x $ |
$ L^\infty_x $ |
$ O(t^{-\frac14}) $ |
$ t $ |
$ +\infty $ |
$ p>7 $ |
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. 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. |
[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. 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. |
[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. |




[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
Other articles
by authors
[Back to Top]