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

HightlightsA broad numerical investigation involving large ensembles of perturbed initial data indicates the Peregrine and second order rational solutions of the NLS equation are linearly unstable.A highly accurate Chebyshev pseudo-spectral method is developed for solving the NLS equation that uses the map x=cot in combination with the Fast Fourier Transform to approximate uxx on an infinite domain.A modified Fourier spectral method that can treat initial data with discontinuous derivatives over periodic domains is developed for solving the NLS equation, of which the resolution of the Peregrine solution is a particular example. The rational solutions of the nonlinear Schrdinger (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 and the FFT to approximate uxx, 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 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.