Deterministic and stochastic surface growth with generalized nonlinearity

The scaling behavior of the interface width for the generalized Kardar-Parisi-Zhang (KPZ) equation {∂}h/{∂}t={ν}∇²h+{λ}{}{∇}h^μ+{η}(x,t) is studied in two dimensions as a function of {μ} with and without the additive noise term {η}. In the case of additive noise, the scaling of the surface width is found for all {μ} to be the same as for the ordinary KPZ equation ({μ}=2) in contrast to a previous conjecture. This appears to be due to the combined action of the nonlinearity {}{∇}h^μ and the noise under renormalization, which together induce a {}{∇}h² term. For the deterministic case corresponding to the smoothing of an initially rough interface with roughness exponent {α}, good agreement with the scaling relation z=min[2,{α}(1-{μ})+{μ}] is obtained for {μ}{≥}1. However, for {μ}1, an instability is observed, which leads to a fluctuating grooved surface. For an asymptotically large system, the roughness exponent is 1 and the growth exponent is approximately equal to 1/2. The evolution of two slightly different surfaces is studied in order to determine a Lyapunov exponent characterizing this instability.