P. Broadbridge et P. Tritscher, AN INTEGRABLE 4TH-ORDER NONLINEAR EVOLUTION EQUATION APPLIED TO THERMAL GROOVING OF METAL-SURFACES, IMA journal of applied mathematics, 53(3), 1994, pp. 249-265
The fourth-order nonlinear partial differential equation for surface d
iffusion is approximated by a new integrable nonlinear evolution equat
ion. Exact solutions are obtained for thermal grooving, subject to bou
ndary conditions representing a section of a grain boundary. When the
slope m of the groove centre is large, the linear model grossly overes
timates the groove depth. In the linear model dimensionless groove dep
th increases linearly with m, but in the nonlinear model it approaches
an upper limit. A nontrivial similarity solution is found for the lim
iting case of a thermal groove whose central slope is vertical.