Similarity solutions to evolution equations in one-dimensional interfaces - Mohammed Benlahsen; Ayman Eldoussouki; Mohammed Guedda; Mustapha Jazar (Electronic Journal of Differential Equations, 2011)
In this note, we study the evolution equation $$ partial_t h = -upartial^2_x h-Kpartial^4_x h +lambda_1(partial_x h)^2-lambda_2partial^2_x(partial_x h)^2. $$ which was introduced by Munoz-Garcia [9] in the context of erosion by ion beam sputtering. We obtain an analytic solution that has the similarity form, which is used in obtaining the coarsening behavior. This solution has amplitude and wavelength that increase like $ln(t)$ and $ sqrt{tln(t)}$, respectively.