Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion

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.