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Summary form only given.The advent of intense femtosecond laser pulses has introduced many challenging propagation related problems, with implications for atmospheric propagation, optical communications, and laser-plasma interactions. In this work, we numerically study the situation where the combination of diffraction, normal dispersion, and cubic nonlinearity lead to pulse splitting. In particular, we focus on effects that lead to recently observed asymmetries in the pulse splitting. In this regime, we find the dynamics are mainly determined by the Raman effect, the shock terms, and the initial third-order temporal phase modulation.