Solutions with expanding compact support of saturated Schrodinger equations: self-similar solutions

We prove the existence of solutions $u(t,x)$ of the Schrodinger equation with a saturation nonlinear term $(u/|u|)$ having compact support, for each $t>0$, that expands with a growth law of the type $C\sqrt{t}$. The primary tool is considering the self-similar solution of the associated equation.