# Roughening in Nonlinear Surface Growth Model
## 1 Introduction  
One of the great challenges of physics and materials science is to understand the growth and surface morphology of the interfaces, both in nature and in technological applications. A recently developed, highly active field of research in statistical physics is dealing with the understanding of surface growth processes [1; 2; 3; 4; 5; 6; 7; 8; 9]. Researchers are challenged to explore the relationship between the structure and properties of nanostructured materials, and to develop nanoscale structures in a conscious and planned way. The industrial application of coating processes allows thin layers with prescribed properties to be formed on a solid substrate [2]. Different layer-forming techniques are used to produce films, such as ion-beam sputtering (IBS), molecular beam epitaxial (MBE), chemical vacuum deposition (CVD) or physical vacuum deposition (PVD). The technique of growth surfaces under MBE has received considerable attention for a wide range of technological and industrial applications. This approach provides unique capability to grow crystalline thin films with precise control of thickness, composition and morphology. This enables scientists to build nanostructures as pyramidal or mound-like objects. The evolution of the surface morphology during MBE growth results from a competition between the molecular flux and the relaxation of the surface profile through surface diffusion. The extraordinary richness of pattern forming during MBE is determined by processes which occur locally at the surface. During the last 30 years, extensive theoretical and experimental research activity has been devoted to this topic. A comprehensive review can be found in [3]. Theoretical models have been proposed which include dynamic scaling [4; 5]. Continuous theory [6], stochastic approaches with nucleation and growth [7], and lattice gauge theory [8] have improved significantly the knowledge of the growth process phenomenon. In surface dynamic evolution, both surface coarseness and smoothening mechanisms are observed, for example, in MBE and PVD [10; 11; 12]. During the MBE, a weak particle board is directed to the sample so that the thin layer can be applied in virtually atomic layers and the composition can beeasily controlled. In the case of vacuum spray, the particles exert a non-thermal effect on the source but generate plasma. During the two processes, the deposition of the particles and the smoothing of the resulting surface patterns are determined by surface diffusion [9; 10; 11; 12; 13; 14; 15]. On the growing surface, self-similar structures can be observed, but the growth of periodic patterns is unstable. In fact, the growth processes of amorphous thin films can be interpreted in an attractive system. Amorphous structures exhibit isotropic spatiality in the absence of a long-term structural sequence. Experimental studies of amorphous thin films, generated by electron beam atomization, show the formation of mound-like structures in mesoscopic length [2; 3; 4; 5; 6; 7; 8; 9; 16; 17]. Stochastic growth equations of continuous continuum models indicate the complexity of atomic size growth processes [1]. Thin films deposited by vacuum techniques such as MBE typically produce thin film of a thickness in the range of 10-25 um. In the working chamber of the MBE equipment, each element is steamed separately in steady state. The speed of vaporization of each element and thus the composition of the layer to be created can be controlled very precisely ([10; 18; 19; 20; 21]). The MBE method has been developed to produce single-crystal layers of semiconductors. The media is usually kept at a few hundred \({}^{\circ}\)C. Atoms arriving on the surface (data meshes) first bind weakly to the surface, migrate and ultimately make a perfect two-dimensional array. Growth speed is relatively slow, usually one monolayer/s.