On a fractional nonlinear Schr\"odinger equation with irregular coefficients. case: d<2s

TL;DR

This paper establishes the existence and uniqueness of very weak solutions for a fractional nonlinear Schrödinger equation with irregular coefficients in dimensions less than twice the fractional order, including numerical observations of solution behaviors.

Contribution

It introduces the concept of very weak solutions for the fractional nonlinear Schrödinger equation with irregular coefficients and proves their well-posedness and compatibility with classical solutions.

Findings

Existence of very weak solutions in the irregular coefficient setting

Uniqueness of these solutions under certain conditions

Numerical experiments revealing interesting solution behaviors

Abstract

In the case when $d<2s$, where $d$ is the space dimension and $s$ is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schr\"odinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: $d=1, s=1$. A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.