# American Institute of Mathematical Sciences

## 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  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, doi: 10.3934/dcdss.2020456
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##### 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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