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| Grain boundaries in extended two-dimensional pattern-forming systems are curves separating regions of slanted rolls. When the angle between the rolls in each of the two regions exceeds a certain threshold, it is known [1,2] that the core of the grain boundary transforms into a chain of convex-concave disclinations. Even though the regularized Cross-Newell (RCN) phase diffusion equation cannot describe this transition all the way to the appearance of defects, it can nevertheless be used to address the question of whether the transition results from an instability of the grain boundary core, and if so, to describe this instability. To this end, we will take full advantage of the existence of an exact grain-boundary solution of RCN and of the variational nature of this equation. I will also show numerical simulations and connect our results to those of Haragus and Scheel [3] on grain boundaries of the Swift-Hohenberg equation.
References:
1. N.M. Ercolani, R. Indik, A.C. Newell, and T. Passot, J. Nonlinear Sci. 10, 223-274 (2000).
2. N.M. Ercolani and S.C. Venkataramani, J. Nonlinear Sci. 19, 267-300 (2009).
3. M. Haragus and A. Scheel, European Journal of Applied Mathematics 23, 737-759 (2012). |
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