Aggradation and degradation in one-dimensional channels are often modeled with a simplified nonlinear diffusion equation. Different degrees of nonlinearity are obtained using the Chezy and Manning/Gauckler-Strickler laws for the friction coefficient combined with a sediment transport equation having a generalized form of the Meyer-Peter and M¿ller formula. Analytical self-similar solutions for the "dam break" and the base-level lowering are presented. While the linear case corresponds to the classic diffusion equation, the main effect of the nonlinearity appears to be the presence of singularities in the self-similar solutions, related to the finite speed of propagation of perturbations.