Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 53, pp. 1-15.

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

Authors: Pascal Begout (Univ. Toulouse Capitole, France)
         Jesus Ildefonso Diaz (Univ. Complutense de Madrid, Spain)

Abstract:
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.

Submitted April 14, 2025. Published May 24, 2025.
Math Subject Classifications: 35C06, 35A01, 35A02, 35J91, 35Q55.
Key Words: Schrodinger equation with saturated nonlinearity; 
 solutions compactly supported; energy method; Dirichlet boundary condition;
 Neumann boundary condition; existence; uniqueness.