We investigate both theoretically and numerically the Riemann pulse (simple Riemann wave) in optical media featuring strong nonlinearity and weak normal higher-order dispersion (HOD). We find that the Riemann pulse under dispersion with arbitrary order $k$ undergoes a progressive steepening process until a shock forms. Employing the semiclassical approximation approach, we derive generalized expressions for the HOD Riemann pulse, whose envelope amplitude is linear to the $k\text{/}2$ power of the pulse chirp. The shock distance and shock time decrease with the dispersion order $k$, while the shock amplitude increases with $k$. Besides, we reveal that loss can delay the shock formation of the HOD Riemann pulse, compared to the lossless case. In the HOD regime, the evolution of the Riemann pulse is significantly influenced by the initial pulse profile and the chirp parameter $C$. Particularly, contrary to the scenario of second-order dispersion, the shock distance, shock time, and shock amplitude of the HOD Riemann pulse exhibit distinct behaviors in the ranges of $C\le 0$ and $C>1$. Theoretical predictions show an excellent agreement with numerical results. Our findings will enrich the fundamental comprehension of shock-wave phenomena in nonlinear optics with complicated dispersion and may inspire potential applications on other physical systems.

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• Dispersion engineering
• Dynamics of nonlinear optical systems
• Ultrafast optics