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

TL;DR

This paper proves the existence of self-similar solutions with expanding compact support for saturated Schrödinger equations, demonstrating a specific growth law and employing self-similarity analysis as the main method.

Contribution

It introduces a new class of solutions with expanding support for saturated Schrödinger equations, using self-similar solutions to establish existence.

Findings

Solutions have support expanding as C√t over time

Existence of solutions with compact support for all t>0

Growth law of the support is explicitly characterized

Abstract

We prove the existence of solutions \(u(t,x)\) of the Schr{\"o}dinger 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. For more information see https://ejde.math.txstate.edu/Volumes/2025/53/abstr.html