The fourth-order nonlinear partial differential equation for surface diffusion is approximated by a new integrable nonlinear evolution equation. Exact solutions are obtained for thermal grooving, subject to boundary conditions representing a section of a grain boundary. When the slope m of the groove centre is large, the linear model grossly overestimates the groove depth. In the linear model dimensionless groove depth increases linearly with m, but in the nonlinear model it approaches an upper limit A nontrivial similarity solution is found for the limiting case of a thermal groove whose central slope is vertical.