Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities.

Since the kinetic and potential energy terms of the real-time nonlinear Schrödinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well-known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schrödinger equation, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high-wave-number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for Deltatkmax2 < or =2pi, where kmax=pi/Deltax.