Numerical investigation of the stability of the rational solutions of the nonlinear Schrodinger equation

The rational solutions of the nonlinear Schrodinger (NLS) equation have been proposed as models for rogue waves. In this article, we develop a highly accurate Chebyshev pseudo spectral method (CPS4) to numerically study the stability of the rational solutions of the NLS equation. The scheme CPS4, using the map x = cot theta and the FFT to approximate u(xx), correctly handles the infinite line problem. A broad numerical investigation using CPS4 and involving large ensembles of perturbed initial data, indicates the Peregrine and second order rational solutions are linearly unstable. Although standard Fourier integrators are often used in current studies of the NLS rational solutions, they do not handle solutions with discontinuous derivatives correctly. Using standard Fourier pseudo-spectral method (FPS4) for Peregrine initial data yields tiny Gibbs oscillations in the first steps of the numerical solution. These oscillations grow to phi(1), providing further evidence of the instability of the Peregrine solution. To resolve the Gibbs oscillations we modify FPS4 using a spectral-splitting technique which significantly improves the numerical solution. (C) 2017 Elsevier Inc. All rights reserved.