Abstract
The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics.
Highlights
Growth patterns in clusters and sodification fronts are challenging problems from a long time
The second term is the lowest-order nonlinear term that can appear in the surface growth equation justified with the Eden model and originates from the tendency of the surface to locally grow normal to itself and has a non-equilibrium in origin
Odor and co-worker intensively examined the two dimensional KPZ equation with extended dynamical simulations to study the physical aging properties of different systems like glasses or polymers [18]. Beyond these continuous models based on partial differential equations (PDEs) there are numerous purely numerical methods available to study diverse surface growth phenomena
Summary
Growth patterns in clusters and sodification fronts are challenging problems from a long time. More general interface growing models were developed based on the so-called Kuramoto-Sivashinsky (KS) equation which is similar to the KPZ model with and extra −∇4u term on the right hand side of (1.1) (see [21], [36]). Odor and co-worker intensively examined the two dimensional KPZ equation with extended dynamical simulations to study the physical aging properties of different systems like glasses or polymers [18]. Beyond these continuous models based on partial differential equations (PDEs) there are numerous purely numerical methods available to study diverse surface growth phenomena. The similarity method is used for the investigation of analytic solution of the two dimensional Navier-Stokes equation with a non-Newtonian type of viscosity [4]
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