Virial functional and dynamics for nonlinear Schrödinger equations of local interactions (Q1705178)

This paper deals with a nonlinear Schrödinger equation of the type \[ i\frac{\partial \psi}{\partial t}+\Delta \psi +f(\psi)=0, \] where the nonlinearity \(f\) has some properties such as superlinearity around zero and real differentiability, and such that conservation of momentum and conservation of mass are implied. In particular, it is known that the solution \(\psi\) satisfies the so called \textit{virial identity} \[ \frac{d^2\;}{dt^2}\int_{\mathbb R^d}|x|^2|\psi(x,t)|^2dx=8\mathcal K(\psi)(t), \] where \(\mathcal K\) is a functional defined as \[ \mathcal K(u)=||\nabla u||^2-\frac d 2 \int_{\mathbb R^d}G(u(x))dx, \] with \(G(z)=\bar{z}f(z)-2F(z)\) and \(2\partial_{\bar z}F=f\). The authors are able to link blow-up and scattering properties of the solution \(\psi\) to the properties of the functional \(\mathcal K\). In particular, they provide three different cases: the solution is uniformly bounded, the solution scatters to a free solution or blows up, the solution exists globally in time.