The fourth-order non-linear boundary value problem for the evolution of a single symmetrical grain boundary groove by surface diffusion was modelled analytically. A solution was obtained by partitioning the surface into sub-intervals which were delimited by lines of constant slope. Within each sub-interval, the advance of the surface was described by an integrable non-linear evolution equation. The model could closely simulate the actual non-linearity. The surface profile was determined for various values of the central groove slope. This included the limiting case of a groove which had a vertical root. Such a solution could exist only because of a non-linearity.

P.Tritscher, P.Broadbridge: Proceedings of the Royal Society A, 1995, 450[1940], 569-87