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Advances in Nonlinear Optics
Volume 2009 (2009), Article ID 926810, 9 pages
Research Article

Localized States in Bi-Pattern Systems

1l'Institut Non Linéaire de Nice (INLN), Centre National de la Recherche Scientifique (CNRS), Université de Nice Sophia Antipolis, 1361 Route des Lucioles, 06560 Valbonne, France
2Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Blanco Encalada 2008, Santiago, Chile
3Instituto de Física, Pontificia Univesidad Católica de Valparaíso, Casilla 4059, Avenida Brasil 2950, Valparaíso, Chile

Received 4 February 2009; Accepted 7 May 2009

Academic Editor: M. Tlidi

Copyright © 2009 U. Bortolozzo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a unifying description of localized states observed in systems with coexistence of two spatially periodic states, called bi-pattern systems. Localized states are pinned over an underlying lattice that is either a self-organized pattern spontaneously generated by the system itself, or a periodic grid created by a spatial forcing. We show that localized states are generic and require only the coexistence of two spatially periodic states. Experimentally, these states have been observed in a nonlinear optical system. At the onset of the spatial bifurcation, a forced one-dimensional amplitude equation is derived for the critical modes, which accounts for the appearance of localized states. By numerical simulations, we show that localized structures persist on two-dimensional systems and exhibit different shapes depending on the symmetry of the supporting patterns.