We study the cubic nonlinear fractional Schrödinger equation with Lévy indices posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp in the sense of index of regularity required for the solutions with respect for known results on the full real line . Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data and the corresponding linear solution. To get our results we use the Colliander and Kenig approach based on the Riemann–Liouville fractional operator combined with Fourier restriction method of Bourgain (1993) and some ideas of the recent work of Erdoğan et al. (2019). The method applies to both focusing and defocusing nonlinearities. As a consequence of our analysis we prove a smoothing effect for the cubic nonlinear fractional Schrödinger equation posed in full line for the case of the low regularity assumption, which was point out at the recent work (Erdoğan et al., 2019).

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半线上的三次非线性分数阶薛定谔方程

我们研究了半直线上带有 Lévy 指数的三次非线性分数阶薛定谔方程。更准确地说，我们定义了该模型的解的概念，并且在考虑完整实数线上的已知结果的解所需的规律性指数意义上，我们获得了几乎尖锐的局部适定性结果。此外，我们证明对于同一模型，非线性部分的解比初始数据和相应的线性解更平滑。为了得到我们的结果，我们使用基于黎曼-刘维尔分数算子的 Colliander 和 Kenig 方法，结合 Bourgain (1993) 的傅里叶限制方法以及 Erdoğan 等人最近工作的一些想法。 （2019）。该方法适用于聚焦和散焦非线性。作为我们分析的结果，我们证明了在低正则性假设的情况下三次非线性分数式薛定谔方程的平滑效应，这一点在最近的工作中指出了（Erdoğan 等人，2019）。