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Abstract. We analyze how the process of bedrock incision by the impact of sediment grains can be described and coupled with sediment dynamics. We first point out that the key parameter is a bedrock dimensionless coefficient that describes the ratio between the volumes of impacting sediments and bedrock erosion. We then write the coupled equations by introducing a partitioning coefficient between sediment and bedrock erosions. It describes the time spent doing one or the other erosion process – or the proportion of depositing grains that impact bedrock. In a 1D along-stream system, the resulting equations lead to a similar description of the cover effect proposed by Sklar and Dietrich (2004), giving a rationale to their expression. In a second step, we extend the concept to lateral erosion or deposition fluxes. We develop analytical solutions for a river fed by uniform lateral sediment fluxes from hillslope and show why the sediment load can exceed the transport capacity. We then implement the equations in the numerical code River.lab/eros, where water depth and velocity, as well as erosion and deposition fluxes, are solved with the method of precipitons. As an example, we simulate the evolution of the Rheinfall at Schaffhausen, Switzerland, a prominent knickpoint along the Hochrhein. In contrast with sediment processes, where the knickpoint slope decreases by diffusion without upstream displacements, bedrock abrasion allows knickpoint to move upstream while retaining almost the same shape. This is consistent with detachment-limited behavior as emphasized in the theoretical part of the paper. The knickpoint shape (foot elevation and height) and retreat rates are highly dependent on the sediment load in river. Bedrock erosion first happens in a narrow canyon that propagates upstream, and then the river widens after the knickpoint passed by.