Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations

Abstract

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