Nonlinear Schrodinger equation

Nonlinear Schrodinger equation (NLS) is in the following form: $$i\frac{du}{dt}=-\Delta_x u + |u|^2u,$$, where $x$ lies in the torus $\mathbb{T}^d$, and $t\in \mathbb{R}$. We are going to study the behavior of the solution $u(t,x)$ ( corresponding to initial value $u(0,x)$). By applying Birkhoff normal forms, we see that in the one dimensional context, all solutions are linear stable. However, in higher dimensions , the answer is not that simple. I will introduce some recent important results, and explain the main idea in each case.