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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

* Corresponding author: serge.dumont@unimes.fr

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
 $p>7$
.
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:

show all references

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