On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

Abstract

We consider a periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = Δu \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1<p\le 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L_{\omega }^1{C}_t{L}_x^2$.

17 pages, comments welcome

References in corpus (1)

• On dispersion managed nonlinear Schrödinger equations with lumped amplification

Cited by in corpus (1)

• Remark on the local well-posedness for NLS with the modulated dispersion