The flow of water in the unsaturated soil is governed by a nonlinear partial differential equation known as the Richards equation. A simplified form of the Richards equation is Burgers equation. In this paper we demonstrate that with a similarity transform, the nonlinear Burgers equation can be mapped to the Bernoulli equation and Riccati equation respectively for different convection terms for the analysis of water flow in soils undergoing surface erosion on hillslopes. The introduction of a dynamic surface in the mathematical model presented in this paper is a more realistic representation of soil water flow under natural flow conditions than other models presently in use. Exact analytical solutions are developed for both the Bernoulli and Riccati equations subject to an initially dry soil. Examples are given with data for real soils under simulated erosion events. The usual Burgers equation is shown to be much easier to use without the need of including the unsaturated hydraulic conductivity.