Members of an hierarchy of integrable nonlinear evolution equations, related to the wellknown linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of the concentration, are adapted to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily-oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. The constitutive relations are explicitly formulated and these show that the theoretical anisotropic material behaves like a liquid crystal. The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material. Closed-form solutions are constructed for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion.
The fourth-order nonlinear boundary-value problem for the evolution of a single symmetric grain-boundary groove by surface diffusion is modelled analytically. A solution is achieved by partitioning the surface into subintervals delimited by lines of constant slope. Within each subinterval, the advance of the surface is described by an integrable nonlinear evolution equation. The model is capable of incorporating the actual nonlinearity arbitrarily closely. The surface profile is determined for various values of the central groove slope including the limiting case of a groove which has a root that is vertical. Such a solution exists only because of the nonlinearity.
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