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

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

Content Provider	MDPI
Author	Duo, Siwei Lakoba, Taras Zhang, Yanzhi
Copyright Year	2021
Abstract	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 ${(- Δ)}^{α / 2}$ . The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power $α \in [1, 2]$ but unstable for $α \in (0, 1)$ . In the focusing case, they can be linearly unstable for any $α \in (0, 2]$ . 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 $α \in (1, 2]$ 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 $α \leq 1$ , 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 $α < 1$ , 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.
Starting Page	1394
e-ISSN	20738994
DOI	10.3390/sym13081394
Journal	Symmetry
Issue Number	8
Volume Number	13
Language	English
Publisher	MDPI
Publisher Date	2021-08-01
Access Restriction	Open
Subject Keyword	Symmetry Mathematical Physics Fractional Nonlinear Schrödinger Equation Fractional Laplacian Plane Wave Solution Modulation Instability Split-step Method Numerical Stability
Content Type	Text
Resource Type	Article

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