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

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 O(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.