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We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditionsfor occurrence of strong anisotropy (SA) in the scaling properties oftwo-dimensional surfaces displaying generic scale invariance. There, a naturalAnsatz was proposed for SA, which arises naturally when analyzing data frome.g. thin-film production experiments. The Ansatz was tested in Gaussian(linear) models of surface dynamics and in non-linear models, like theHwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813 (1989)], which aresusceptible of accurate approximations through the former. In contrast, here weanalyze non-linear equations for which such type of approximations fail.Working within generically-scale-invariant situations, and as representativecase studies, we formulate and study a generalization of the HK equation forconserved dynamics, and reconsider well-known systems, such as the conservedand the non-conserved anisotropic Kardar-Parisi-Zhang equations. Through thecombined use of Dynamic Renormalization Group analysis and direct numericalsimulations, we conclude that the occurrence of SA in two-dimensional surfacesrequires dynamics to be conserved. We find that, moreover, SA is not generic inparameter space but requires, rather, specific shapes from the terms appearingin the equation of motion, whose justification needs detailed information onthe dynamical process that is being modeled in each particular case.

Strong anisotropy in two-dimensional surfaces with generic scale invariance: Non-linear effects