Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

Jean-Baptiste Casteras, Juraj Foldes, Gennady Uraltsev

Published 2022-03-07Version 1

In this paper, we study the local well-posedness of the cubic Schr\" odinger equation: $$ (i\partial_t - P) u = \pm |u|^2 u\quad \textrm{ on }\ I\times \R^d ,$$ with randomized initial data, and $P$ being an operator of degree $s \geq 2$. Using careful estimates in anisotropic spaces, we improve and extend known results for the standard Schr\"odinger equation (that is, $P$ being Laplacian) to any dimension under natural assumptions on $P$, whose Fourier symbol might be sign changing. Quite interestingly, we also exhibit the existence of a new regime depending on $s$ and $d$, which was not present for the Laplacian.