An Inhomogeneous, $L^2$-Critical, Nonlinear Schrödinger Equation

Abstract

An inhomogeneous nonlinear Schrödinger equation is considered, which is invariant under $L^2$-scaling. The sharp condition for global existence of $H^1$-solutions is established, involving the$L^2$-norm of the ground state of the stationary equation. Strong instability of standing waves is proved by constructing self-similar solutions blowing up in finite time.