On the Numerical Study of the Rational Solutions of the Nonlinear Schrödinger Equation.

The stability of the rational solutions of the nonlinear Schr odinger (NLS) equation has only recently began to be addressed. In this paper we develop a Chebyshev pseudo-spectral method for the NLS equation to study the stability of the rational solutions. Using the map x = cot θ and the Fast Fourier Transform (FFT) to approximate uxx, the Chebyshev scheme effectively handles the infinite line boundary conditions. An extensive numerical study, involving large ensembles of perturbed initial data for the Peregrine solution (the lowest order rational solution), indicates it is linearly unstable. Working with unstable solutions is numerically challenging. In the current literature, numerical experiments related to the Peregrine solution frequently employ standard Fourier methods without a discussion of the related numerical issues. We examine the performance of a Fourier pseudo-spectral method (FPS4) using Peregrine initial data. Applying FPS4, tiny Gibbs oscillations occur in the first few steps of the numerical solution. These oscillations grow to O(1), providing further evidence of the instability of the Peregrine solution. We modify the FPS4 method using a spectral-splitting technique which resolves the Gibbs oscillations and significantly improves the numerical solution. [ABSTRACT FROM AUTHOR]