# Roughening in Nonlinear Surface Growth Model
## 2 Physical Background  
In the mathematical approach, it is very important to incorporate the uncertainties of the parameters into the model. The main sources of uncertainties are difficult to predict. These are for example, the atomic level of elastic conjoint effects and surface state changes. Partial differential equations describing the characterization of irregular surfaces are provided with free boundary conditions. Partial differential equations cannot be solved analytically or only very rarely, and the numerical algorithms used are generally unstable. Therefore, variational methods are required. Governing equations describing the phenomenon are usually strongly nonlinear differential equations. In the surface coating processes, the initial state of the substrate is an almost flat surface, the particles in the vapor are perpendicular to the surface and the deposition process is characterized by the deposition flux. Particles on the surface go through various surface diffusion processes when they reach their final position. The growing surface layer is composed of these particles after a certain time. The height or morphology \(H(\mathbf{t},\;\mathbf{x},\;\mathbf{y})\) of the surface layer depends on time \(t\) and spatial coordinates \(x,y\). The evaluation of the surface structure is determined by the particles arriving on the surface and by the condensed particles on the surface. The process in time and space is determined by the interplay of different mechanisms such as coarseness, smoothening and shape formation. To examine the roughness of the surface, we control the height profile with \(h(\mathbf{t},\;\mathbf{x},\;\mathbf{y})=H(\mathbf{t},\;\mathbf{x},\;\mathbf{y })-Ft\), where \(F\) is the mean deposition rate and \(h\) satisfies the equation:  
\[h_{t}=G(\
abla h)\]  
with a functional \(G\) of spatial derivatives and depending on the setting parameters of the deposition procedure and material parameters. Our goal is to investigate deterministic equation whose solutions are properly characterizing the physical phenomenon and from which the results obtained with the initial condition are likely to remain valid even after a long period of time. The equations are sometimes supplemented with stochastic members that represent temperature or instrumental noises. In this paper we do not address the effects of noise. The processes in time \(t\) are analysed in one dimension by one spatial coordinate \(x\). The surface height is characterized by the height function \(h(t,x)\) from the theoretical plane surface. For amorphous growth processes temporal and translational invariance in growth direction and in the direction orthogonal to growth are considered [3]. The isotropy of the amorphous phase implies invariance under rotation and reflection around the direction of growth. Therefore, the presence of the derivatives terms of odd order of \(h(t,x)\) is not allowed. The symmetry up and down (\(h\rightarrow-h\)) is necessary. The dynamics can be described by effective one-dimensional equations. The simplest equation describing the film growth is the Edward-Wilkinson model [18]:  
\[h_{t}=v\;h_{xx}+\eta, \tag{1}\]  
where the indices denote the derivative with respect to the variable \(t\) or \(x\), and \(v\) is constant and \(\eta\) denotes the noise term. The term \(v\;h_{xx}\) is the surface tension. For example, when desorption is important, the growth can be described by the Kardar-Parisi-Zhang equation [4] which is considered as the paradigm of the stochastic roughening. For the analytical treatment of the vapor phase deposition of thin films, a nonlinear term was added to Equation (1) to model the deposition and formation of a thin film:  
\[h_{t}=v\;h_{xx}+\lambda\;(h_{x})^{2}+\eta, \tag{2}\]  
where \(\lambda\) is a constant. In Equation (2), \(h_{t}\) represents the rate of absorption, the first term at the left side the relaxation of the interface and the second term the tendency of the surface locally growing normal to itself.