We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian ${(-\Delta )}^{\alpha /2}$. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power $\alpha \in [1,2]$ but unstable for $\alpha \in (0,1)$. In the focusing case, they can be linearly unstable for any $\alpha \in (0,2]$. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as $\alpha \in (1,2]$ decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For $\alpha \le 1$, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for $\alpha <1$, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities.

中文翻译：

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具有长程色散的分数阶非线性薛定谔方程中的平面波动力学

我们通过分析和数值研究了分数非线性薛定谔 (NLS) 方程的平面波解的稳定性和动力学，其中远程色散由分数拉普拉斯算子描述 ${(-\Delta )}^{\alpha /2}$. 线性稳定性分析表明，散焦 NLS 中的平面波解总是稳定的，如果功率$\alpha \in [1,2]$ 但不稳定 $\alpha \in (0,1)$. 在聚焦的情况下，它们可能是线性不稳定的$\alpha \in (0,2]$. 然后，我们应用分步傅立叶光谱 (SSFS) 方法来模拟平面波动力学的非线性阶段。与早期对分数聚焦 NLS 的孤立波解的研究一致，我们发现作为$\alpha \in (1,2]$减小，解决方案演变为存在于小幅度色散波“海”背景上的越来越局部化的脉冲。这种高度局部化的脉冲具有很宽的频谱，其大部分模式在脉冲演化的非线性阶段被激发，并且无法通过线性稳定性分析预测。为了$\alpha \le 1$，我们总能找到崩溃的解决方案。据我们所知，我们还首次证明，对于具有非零群速度（行进平面波）的初始条件，与驻平面波初始条件相比，坍缩的开始被延迟。对于散焦分数 NLS，即使我们发现行进的平面波线性不稳定$\alpha <1$，我们从未观察到崩溃。作为我们数值研究的副产品，我们推导出了 SSFS 时间步长的稳定性条件，以保证该方法没有数值不稳定性。