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

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