Debris flows often exhibit coherent wave structures—shocklike roll waves on steeper slopes and weaker, more sinusoidal dispersive pulses on gentler slopes. Coarse-rich heads raise basal resistance, whereas fines-rich tails lower it; in gentle reaches, small-amplitude pulses can locally transport momentum across low-resistance segments. We focus on this gentle-slope, long-wave, low-amplitude regime, where the base-flow Froude number is order unity. In this limit, we obtain a Korteweg–de Vries (KdV) reduction from depth-averaged balances with frictional (Coulomb) and viscous-plastic basal options, using a curvature-type internal normal-stress closure in the long-wave small-$k$ regime. Multiple-scale analysis yields effective nonlinear and dispersive coefficients. We also introduce a practical nonlinearity diagnostic that can be computed from observed crest speeds and flow thicknesses. When laboratory-frame crest celerity is available, we estimate an effective quadratic coefficient from the KdV speed-amplitude relation and report its ratio to the shallow-water reference. When only a depth-averaged first-surge speed and thickness are available, we use the same construction to form a velocity-based proxy and note its bias near zero. A Froude-slope diagram organizes published cases into a steep-slope roll-wave domain and a gentle-slope corridor where KdV pulses are admissible. Numerical solutions of the full depth-averaged model produce cnoidal and solitary waves that agree with the reduced KdV predictions within this corridor. We regard dispersive pulses as a regime-specific complement to roll-wave dynamics, offering a condition-dependent contribution to mobility on gentle reaches rather than a universal explanation for long runout.

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• Geophysical fluid dynamics
• Granular flows
• Low Reynolds number flows
• Multiphase flows
• Nonlinear dynamics in fluids