Solutions with expanding compact support of saturated Schr{ö}dinger equations: self-similar solutions

We prove the existence of solutions $u(t,x)$ of the Schr{ö}dinger equation with a saturation nonlinear term $(u\mathrm{/}\mathrm{\mid }u\mathrm{\mid })$ 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. For more information see https://ejde.math.txstate.edu/Volumes/2025/53/abstr.html