In this work, we derive a novel local evolution equation for the interface height, $h(x,t)$, in the context of erosion by ion beam sputtering 1.
This equation is the 1D counterpart of the height equations obtained in Ref.
2. In order to reduce the number of parameters and simplify its analysis, we can properly rescale it to obtain the following single-parameter
equation:
equation
\partial_t h(x,t)= - \partial_x^2 h - \partial_x^4 h + (\partial_x h)^2- r \partial_x^2(\partial_x h)^2,
equation
where $r$ is the squared ratio of a linear crossover length scale to a nonlinear crossover length scale.
The rescaled single parameter equation interpolates between the
Kuramoto-Sivashinsky (KS) equation 3, which presents a chaotic solution and no
coarsening, and the ``conserved'' KS equation 4, which displays unbounded
coarsening. We present numerical simulations of this equation which show
interrupted coarsening in which an ordered cell pattern develops with constant
wavelength and amplitude at intermediate distances, while the profile is
disordered and rough at larger distances. Moreover, for a wide range of
parameters the lateral extent of ordered domains ranges up to tens of cells, in
contrast to others equations which also present invariant under global height
shifts, $h(x, t) h(x, t)+const.$ 5. This
effective interface equation seems to provide an instance of a system with local
interactions in which, for appropriate parameter values, a pattern is stabilized
with constant wavelength and amplitude and introduces a novel scenario in the
context of pattern formation for interfacial evolution equations with these
symmetries 6,7. We complete our numerical study with analytical
estimates for the stationary pattern wavelength and the mean growth velocity for large values of $r$.
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1) Javier Muñoz-Garc\'ıa, Rodolfo Cuerno, and Mario Castro. Phys. Rev. E 74,
050103(R) (2006).\\
2) M. Castro, R. Cuerno, L. Vázquez y R. Gago, Phys. Rev. Lett. 94,
016102 (2005); J. Muñoz-Garc\'ıa, M. Castro y R. Cuerno, Phys. Rev. Lett.
96, 086101 (2006).\\
3) G. I. Sivashinsky, Annu. Rev. Fluid Mech. 15, 179 (1983); Y. Kuramoto,
Chemical Oscillation, Waves and Turbulence. Springer, Berlin, 1984.\\
4) M. Raible, S. J. Linz, and P. Hänggi, Phys. Rev. E 62, 1691
(2000); T. Frisch and A. Verga, Phys. Rev. Lett. 96, 166104 (2006); P. Politi
and C. Misbah, Phys. Rev. E 75, 027202 (2007).\\
5) M. Castro, J. Muñoz-Garc\'ıa, R. Cuerno, M. Garc\'ıa-Hernández, and L. Vázquez. To be published in New. J. Phys. (2007).\\
6) J. Krug, Adv. Compl. Sys. 4, 353 (2001).\\
7) P. Politi and C. Misbah, Phys. Rev. Lett. 92, 090601 (2004); Phys. Rev.
E 73, 036133 (2006).