%0 Book Section %1 statphys23_0799 %A noz garc\'ıa, J. Mu\ %A Cuerno, R. %A Castro, M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K coarsening equations evolution formation interfacial interrupted ion-beam kinetic pattern roughening sputtering statphys23 topic-4 %T Short-range stationary patterns and long-range disorder in an evolution equation for one-dimensional interfaces %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=799 %X 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 $r$. \bigskip 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).

@incollection{statphys23_0799, abstract = {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: \begin{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, \end{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, $h(x, t) \rightarrow h(x, t)+const.$ [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 $r$. \bigskip 1) Javier Mu\~noz-Garc\'{\i}a, Rodolfo Cuerno, and Mario Castro. Phys. Rev. E 74, 050103(R) (2006).\\ 2) M. Castro, R. Cuerno, L. V\'azquez y R. Gago, Phys. Rev. Lett. 94, 016102 (2005); J. Mu\~noz-Garc\'{\i}a, 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. Mu\~noz-Garc\'{\i}a, R. Cuerno, M. Garc\'{\i}a-Hern\'andez, and L. V\'azquez. 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).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {noz \uppercase{g}arc\'{\i}a, J. Mu\ and Cuerno, R. and Castro, M.}, biburl = {https://www.bibsonomy.org/bibtex/270c95bb172a017032250f7a5fd62824b/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {f7795a4c353ecb6be33938616958d6c3}, intrahash = {70c95bb172a017032250f7a5fd62824b}, keywords = {coarsening equations evolution formation interfacial interrupted ion-beam kinetic pattern roughening sputtering statphys23 topic-4}, month = {9-13 July}, timestamp = {2007-06-20T10:16:29.000+0200}, title = {Short-range stationary patterns and long-range disorder in an evolution equation for one-dimensional interfaces}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=799}, year = 2007 }