Comprehensive Analysis of a Stochastic Continuum Equation of Ion-Sputtered Nanostructured Surfaces via Symmetries

For some time now a great deal of interest has been generated by the formation of ordered patterns ofnanostructures on material surfaces, particularly under the conditions that the surfaces are bombarded by a beam of ions,towards development in nanotechnology. In this paper, we have comprehensively analysed the nonlinear continuummodel proposed by Cuerno and Barabasi via symmetries. This stochastic partial differential equation (SPDE) can beregarded as the most significant and widely acceptable theoretical description of oblique incidence ion sputterednanostructured surfaces until now. For this purpose, the symmetry group of the equation is determined and some resultsrelated to the algebraic structure of the Lie algebra of symmetries are given. Also, the optimal system of subalgebraswhich provides the preliminary classification of group invariant solutions is constructed. Mainly, the group invariantsolutions and the similarity reduced equations associated to the infinitesimal symmetries are obtained. Meanwhile, it isnoticeable that this procedure results new solutions of this well-known stochastic equation.

Keywords:

Nanostructures , Stochastic Partial Differential Equations (SPDE) , Lie Symmetry Method , Nano-patternFormation , Invariant Solutions , Optimal System