On the well-posedness of the periodic fractional Schr\"{o}dinger equation

TL;DR

This paper investigates the local well-posedness of the periodic fractional nonlinear Schrödinger equation with various nonlinearities, establishing conditions under which solutions exist, are unique, and depend continuously on initial data.

Contribution

It provides new results on the well-posedness of fractional Schrödinger equations with complex nonlinearities on periodic domains, including cases with logarithmic and power-type nonlinearities.

Findings

Established local well-posedness for initial data with non-vanishing condition.

Analyzed equations with nonlinearities like logarithmic and power sums.

Utilized properties of periodic Sobolev spaces to prove results.

Abstract

We consider the periodic fractional nonlinear Schr\"{o}dinger equation $$ iu_t -(-\Delta)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two ways: the first one $\mathcal{N}\in C^J(\mathbb R^+)$, whose derivatives have a certain polynomial decay, e.g., $\mathcal{N}(|u|)=\log(|u|)$; the second one is given by a sum of powers, possibly infinite, $$ \mathcal{N}(|u|) = \sum a_k |u|^{\gamma_k}, \quad \gamma_k \in \mathbb{R}, ~~ a_k \in \mathbb{C}, $$ which includes examples such as $\mathcal{N}(|u|) \, u =\frac{u}{|u|^{\gamma}},$ $\gamma>0$. By using standard properties of periodic Sobolev spaces $H^J(\mathbb{T}^N)$, $J>0$, we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfy a non-vanishing condition $\inf\limits_{x\in \mathbb{T}^N}|u_0(x)|>0$.