Non-linear kinematic wave for debris flow can be written as follows with complex non-linear coefficients:
where  is the debris flow height,  is the time,  is the downstream channel position,  is the pressure gradient and the depth dependent nonlinear variable wave speed, and  is a flow height and pressure gradient dependent variable diffusion term.
This equation can also be written in the conservative form:
where  is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For  this equation reduces to the Burgers' equation.