An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

The fourth-order nonlinear partial differential equation forsurface diffusion is approximated by a new integrable nonlinearevolution equation. Exact solutions are obtained for thermalgrooving, subject to boundary conditions representing a sectionof a grain boundary. When the slope m of the groove centre islarge, the linear model grossly overestimates the groove depth.In the linear model dimensionless groove depth increases linearlywith m, but in the nonlinear model it approaches an upper limitA nontrivial similarity solution is found for the limiting caseof a thermal groove whose central slope is vertical.