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Since the kinetic and the potential energy term of the real time nonlinearSchr\"odinger equation can each be solved exactly, the entire equation can besolved to any order via splitting algorithms. We verified the fourth-orderconvergence of some well known algorithms by solving the Gross-Pitaevskiiequation numerically. All such splitting algorithms suffer from a latentnumerical instability even when the total energy is very well conserved. Adetail error analysis reveals that the noise, or elementary excitations of thenonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability isdue to the exponential growth of high wave number noises caused by thesplitting process. For a continuum wave function, this instability isunavoidable no matter how small the time step. For a discrete wave function,the instability can be avoided only for $\dt k_{max}^2{<\atop\sim}2 \pi$, where$k_{max}=\pi/\Delta x$.

Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities