In the present paper, the authors develop and analyze new numerical method for the integrable nonlinear Schrödinger equation and the nonlinear Korteweg-de Vries equation with step function initial data and periodic boundary conditions. The method is based on the operator splitting strategy, which also highlights the interplay between the linear and nonlinear parts of the equation. The authors study the convergence and stability properties and prove convergence for the nonlinear Schrödinger equation. On the other hand they explain where difficulties in the proof of convergence for the Korteweg-de Vries equation arise. They also found that the effects of dispersive quantization and fractalization persist into the nonlinear regime. However, it is not clear whether the small oscillations appearing between the jumps are due to numerical error or the persistence of a small fractal contribution.