Strong anisotropy in two-dimensional surfaces with generic scale invariance: nonlinear effects.

We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy in the scaling properties of two-dimensional surfaces displaying generic scale invariance. In that study, a natural scaling ansatz was proposed for strongly anisotropic systems, which arises naturally when analyzing data from, e.g., thin-film production experiments. The ansatz was tested in Gaussian (linear) models of surface dynamics and in nonlinear models, like the Hwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813 (1989)], which are susceptible of accurate approximations through the former. In contrast, here we analyze nonlinear equations for which such approximations fail. Working within generically scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics and reconsider well-known systems, such as the conserved and the nonconserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of dynamic renormalization group analysis and direct numerical simulations, we conclude that the occurrence of strong anisotropy in two-dimensional surfaces requires dynamics to be conserved. We find that, moreover, strong anisotropy is not generic in parameter space but requires, rather, specific forms of the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.

Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models.

Among systems that display generic scale invariance, those whose asymptotic properties are anisotropic in space (strong anisotropy, SA) have received relatively less attention, especially in the context of kinetic roughening for two-dimensional surfaces. This is in contrast with their experimental ubiquity, e.g., in the context of thin-film production by diverse techniques. Based on exact results for integrable (linear) cases, here we formulate a SA ansatz that, albeit equivalent to existing ones borrowed from equilibrium critical phenomena, is more naturally adapted to the type of observables that are measured in experiments on the dynamics of thin films, such as one- and two-dimensional height structure factors. We test our ansatz on a paradigmatic nonlinear stochastic equation displaying strong anisotropy like the Hwa-Kardar equation [Phys. Rev. Lett. 62, 1813 (1989)], which was initially proposed to describe the interface dynamics of running sand piles. A very important role to elucidate its SA properties is played by an accurate (Gaussian) approximation through a nonlocal linear equation that shares the same asymptotic properties.

Kardar-Parisi-Zhang asymptotics for the two-dimensional noisy Kuramoto-Sivashinsky equation.

We study numerically the Kuramoto-Sivashinsky equation forced by external white noise in two space dimensions, that is a generic model for, e.g., surface kinetic roughening in the presence of morphological instabilities. Large scale simulations using a pseudospectral numerical scheme allow us to retrieve Kardar-Parisi-Zhang (KPZ) scaling as the asymptotic state of the system, as in the one-dimensional (1D) case. However, this is only the case for sufficiently large values of the coupling and/or system size, so that previous conclusions on non-KPZ asymptotics are demonstrated as finite size effects. Crossover effects are comparatively stronger for the two-dimensional case than for the 1D system.