Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations

  • Frank Merle

Abstract

We review qualitative properties of solutions of critical nonlinear Schrödinger equation iut=Δuup1u,u(0)=u0,iu_t= -\Delta u-| u|^{p-1}u, \quad u(0)= u_0, and Zakharov equations iut=Δu+nu,nt=v,1c02vt=(n+u2),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.