TY - JOUR
T1 - The phase structure of grain boundaries
AU - Ercolani, Nicholas M.
AU - Kamburov, Nikola
AU - Lega, Joceline
N1 - Funding Information:
Data accessibility. This article has no additional data. Authors’ contributions. J.L. designed and performed the numerical explorations, and wrote a first draft of the manuscript. N.M.E. provided expertise in the analysis of the RCN equation, including theorem 1.1 and its proof, as well as background on the Hilbert transform. N.K. proved uniqueness of the minimizer of theorem 1.1. All authors discussed the simulations and the results of this study and contributed to the writing of the final article, which they also read and approved. Competing interests. The authors declare that they have no competing interests. Funding. This material is based upon the work supported by the National Science Foundation under Grant No. DMS-1615921. Acknowledgements. J.L. thanks Arnd Scheel for discussing progress on the work described in [23] and for sharing their manuscript before its publication. N.M.E. and J.L. acknowledge useful discussions with Paul Carter, Gabriela Jaramillo and Lidia Mrad.
Funding Information:
This material is based upon the work supported by the National Science Foundation under Grant No. DMS-1615921.
Publisher Copyright:
© 2018 The Author(s) Published by the Royal Society. All rights reserved.
PY - 2018/4/13
Y1 - 2018/4/13
N2 - This article discusses numerical and analytical results on grain boundaries, which are line defects that separate roll patterns oriented in different directions. The work is set in the context of a canonical pattern-forming system, the Swift-Hohenberg (SH) equation, and of its phase diffusion equation, the regularized Cross-Newell equation. It is well known that, as the angle made by the rolls on each side of a grain boundary is decreased, dislocations appear at the core of the defect. Our goal is to shed some light on this transition, which provides an example of defect formation in a system that is variational. Numerical results of the SH equation that aim to analyse the phase structure of far-from-threshold grain boundaries are presented. These observations are then connected to properties of the associated phase diffusion equation. Outcomes of this work regarding the role played by phase derivatives in the creation of defects in pattern-forming systems, about the role of harmonic analysis in understanding the phase structure in such systems, and future research directions are also discussed. This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.
AB - This article discusses numerical and analytical results on grain boundaries, which are line defects that separate roll patterns oriented in different directions. The work is set in the context of a canonical pattern-forming system, the Swift-Hohenberg (SH) equation, and of its phase diffusion equation, the regularized Cross-Newell equation. It is well known that, as the angle made by the rolls on each side of a grain boundary is decreased, dislocations appear at the core of the defect. Our goal is to shed some light on this transition, which provides an example of defect formation in a system that is variational. Numerical results of the SH equation that aim to analyse the phase structure of far-from-threshold grain boundaries are presented. These observations are then connected to properties of the associated phase diffusion equation. Outcomes of this work regarding the role played by phase derivatives in the creation of defects in pattern-forming systems, about the role of harmonic analysis in understanding the phase structure in such systems, and future research directions are also discussed. This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.
KW - Cross-Newell equation
KW - Defects
KW - Grain boundaries
KW - Pattern-forming system
KW - Swift-Hohenberg equation
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U2 - 10.1098/rsta.2017.0193
DO - 10.1098/rsta.2017.0193
M3 - Article
C2 - 29507177
AN - SCOPUS:85045527162
SN - 1364-503X
VL - 376
JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2117
M1 - 20170193
ER -