Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion

Serge Dumont; Olivier Goubet; Youcef Mammeri

doi:10.3934/dcdss.2020456

Article Contents

2021, Volume 14, Issue 8: 2877-2891. Doi: 10.3934/dcdss.2020456

This issue Previous Article The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data Next Article An extension of the landweber regularization for a backward time fractional wave problem

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
^* Corresponding author: serge.dumont@unimes.fr

This article is dedicated to the memory of Ezzeddine Zahrouni.

Received: February 2020

Revised: September 2020

Early access: November 2020

Published: August 2021

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Abstract

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 $.

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Mathematics Subject Classification: Primary:35Q55;Secondary:65C30, 60H35.

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Figure 1. Graphical representation of the function $ f $ (-) and the convex upper bound $ g $ (- - -)

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

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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)

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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)

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

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