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

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

    * Corresponding author: serge.dumont@unimes.fr 

This article is dedicated to the memory of Ezzeddine Zahrouni.

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  • 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 $.

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


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Graphical representation of the function $ f $ (-) and the convex upper bound $ g $ (- - -)

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

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

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