Nonlinear singular perturbations of the fractional Schr\"odinger equation in dimension one

TL;DR

This paper studies the effects of nonlinear delta-type perturbations on the fractional Schrödinger equation in one dimension, analyzing well-posedness, conservation laws, blow-up solutions, and standing waves for fractional powers between 0.5 and 1.

Contribution

It introduces a detailed analysis of nonlinear singular delta perturbations in the fractional Schrödinger equation, including well-posedness and solution behaviors for fractional orders in (0.5, 1].

Findings

Established local and global well-posedness results.

Identified conditions for blow-up solutions.

Analyzed existence of standing waves.

Abstract

The paper discusses nonlinear singular perturbations of delta type of the fractional Schr\"odinger equation $\imath\partial_t\psi=\left(-\triangle\right)^s\psi$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.