Article Contents
Article Contents

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

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

• 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.

 Citation:

• 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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