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

Abstract

<p style='text-indent:20px;'>In this article, the asymptotic behavior of the solution to the following one dimensional Schrödinger equations with white noise dispersion <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ idu + u_{xx}\circ dW+ |u|^{p-1}udt = 0 $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>is studied. Here the equation is written in the { Stratonovich} formulation, and <inline-formula><tex-math id="M1">\begin{document}$ W(t) $\end{document}</tex-math></inline-formula> is a standard real valued Brownian motion. After establishing the global well-posedness, theoretical proof and numerical investigations are provided showing that, for a deterministic small enough initial data in <inline-formula><tex-math id="M2">\begin{document}$ L^1_x\cap H^1_x $\end{document}</tex-math></inline-formula>, the expectation of the <inline-formula><tex-math id="M3">\begin{document}$ L^\infty_x $\end{document}</tex-math></inline-formula> norm of the solutions decay to zero at <inline-formula><tex-math id="M4">\begin{document}$ O(t^{-\frac14}) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>, as soon as <inline-formula><tex-math id="M7">\begin{document}$ p&gt;7 $\end{document}</tex-math></inline-formula>.