Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation (Q1325808)

The nonlinear Schrödinger equation \[ iu_ t+ \Delta u+ \varepsilon| u|^ \alpha u=0, \qquad x\in\mathbb{R}^ n \tag{1} \] is investigated. A solution \(u\) of (1) is self-similar if \(u(x,t)= \lambda^{-i\omega+ 2/\alpha} u(\lambda^ 2 t,\lambda t)\) for all \(\lambda>0\) and some fixed \(\omega\in \mathbb{R}\). In this paper the critical power \(\alpha= 4/n\) is considered.