Erosion by bedrock river channels is commonly modeled with the stream power equation. We present a two-part approach to solving this nonlinear equation analytically and explore the implications for evolving river profiles. First, a method for non-dimensionalizing the stream power equation transforms river profiles in steady state with respect to uniform uplift into a straight line in dimensionless distance-elevation space. Second, a method that tracks the upstream migration of slope patches, which are mathematical entities that carry information about downstream river states, provides a basis for constructing analytical solutions. Slope patch analysis explains why the transient morphology of dimensionless river profiles differs fundamentally if the exponent on channel slope, n, is less than or greater than one and why only concave-up migrating knickpoints persist when n < 1, whereas only concave-down migrating knickpoints persist when n > 1. At migrating knickpoints, slope patches and the information they carry are lost, a phenomenon that fundamentally limits the potential for reconstructing tectonic histories from bedrock river profiles. Stationary knickpoints, which can arise from spatially varying uplift rates, differ from migrating knickpoints in that slope patches and the information they carry are not lost. Counterparts to migrating knickpoints, called “stretch zones,” are created when closely spaced slope patches spread to form smooth curves in distance-elevation space. These theoretical results are illustrated with examples from the California King Range and the Central Apennines.