Fractional nonlinear Schr\"odinger and Hartree equations in modulation spaces

TL;DR

This paper proves global well-posedness for a class of fractional nonlinear Schrödinger equations with radial initial data in modulation spaces, covering power-type and Hartree-type nonlinearities, for certain dispersion orders.

Contribution

It establishes well-posedness results in modulation spaces for fractional Schrödinger equations with specific nonlinearities and dispersion parameters, extending previous understanding.

Findings

Global well-posedness in modulation spaces for fractional Schrödinger equations.

Applicable to power-type and Hartree-type nonlinearities.

Results hold for dispersion order between rac{2n}{2n-1}rac{2n}{2n-1} and 2.

Abstract

We establish global well-posedness for the mass subcritical nonlinear fractional Schr\"odinger equation $$iu_t - (-\Delta)^\frac{\beta}{2} u+F(u)=0$$ having radial initial data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R^n)$ for $n \geq 2, p>2$ and $p$ sufficiently close to $2.$ The nonlinearity $F(u)$ is either of power-type $F(u)=\pm (|u|^{\alpha}u)\; (0<\alpha<2\beta / n)$ or Hartree-type $(|x|^{-\nu} \ast |u|^{2})u \; (0<\nu<\min\{\beta,n\}).$ Our order of dispersion $\beta$ lies in $(2n/ (2n-1), 2).$