Scaling properties of a class of interfacial singular equations

This paper can be considered as an introductory review of scale invariance theories illustrated by the study of the equation ∂th=−∂x∂xh1−2ν+∂xxxh, where ν>1/2. The d−dimensionals version of this equation is proposed for ν≥1 to discuss the coarsening of growing interfaces that induce a mound-type structure without slope selection (Golubović, 1997). Firstly, the above equation is investigated in detail by using a dynamic scaling approach, thus allowing for obtaining a wide range of dynamic scaling functions (or pseudosimilarity solutions) which lend themselves to similarity properties. In addition, it is shown that these similarity solutions are spatial periodic solutions for any ν>1/2, confirming that the interfacial equation undergoes a perpetual coarsening process. The exponents β and α describing, respectively, the growth laws of the interfacial width and the mound lateral size are found to be exactly β=(1+ν)/4ν and α=1/4, for any ν>12. Our analytical contribution examines the scaling analysis in detail and exhibits the geometrical properties of the profile or scaling functions. Our finding coincides with the result previously presented by Golubović for 0<ν≤3/2.