The Korteweg-de Vries equation is a famous model for the propagation of long waves in a shallow canal. In generalization of this model with stronger nonlinear effects a competition between dispersion and anti-diffusion is possible. Solutions to these equations can fail to depend continuously on data unless data has extra decay. In this talk, joint work with David Ambrose and Doug Wright, we investigate a wide class of equations, where this extra decay is not needed.