Stochastic Dynamics and Selection in the One Dimensional Stabilized Kuramoto-Sivashinsky Equation

Many spatially extended nonlinear systems are known to exhibit coarsening - starting from an initial uniform (disordered) state, ordered structures appear, the size of which increases with time. We study coarsening dynamics in the stabilized Kuramoto-Sivashinsky (SKS) equation in one dimension, with and without noise. The SKS equation is used to describe the growth of crystal surfaces, in particular the motion of terrace edges in step-flow growth [1]. The key feature of this equation is that it displays a bifurcation from a uniform steady state to a band of periodic states, depending on the control parameter. Coarsening is studied by analyzing the time evolution of the structure function for a range of control parameter values, starting from a uniform initial state. We find that the width of the structure function decays as a power law with time during an intermediate time regime, until a narrow peak centered at a given wave number is obtained. This is consistent with the emergence of an ordered (periodic) state which grows in spatial extent. We calculate the decay exponents and discuss the influence of the noise amplitude on the values of the exponents. We also make connections with wave number selection.

[1] I. Bena et al, Phys. Rev. B, 47, 7408 (1993).