Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations

Abstract

We review qualitative properties of solutions of critical nonlinear Schrödinger equation $i{u}_t=-\Delta u-|u{|}^{p-1}u,u(0)=u_0,$ and Zakharov equations $i{u}_t=-\Delta u+nu,n_t=-\nabla \cdot v,\frac{1}{c_0^2}{v}_t=-\nabla (n+|u{|}^2),$ which develop a singularity in finite time.