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Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

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Abstract

We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the shadow kink. Its local profile is described by the generalized Hastings and McLeod solutions of the second Painlevé equation (Claeys et al. in Ann Math 168(2):601–641, 2008; Hastings and McLeod in Arch Ration Mech Anal 73(1):31–51, 1980). As part of our analysis we give a new proof of existence of these solutions.

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Notes

  1. By changing y by \(-y\), we obtain the solutions of (1.21) corresponding to \(\alpha \ge 0\).

  2. By differentiating (3.5) we can also obtain the boundedness of \({\tilde{v}}'''\) on compact intervals (provided \(f\in C^1(\mathbb {R})\)). Then, the convergence in Theorem 1.2 can be improved to \(C^2\) convergence on compacts.

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Acknowledgements

We would like to thank William Troy and Stuart Hastings for observations that helped us to implement some important improvements in the present version of this work. We would like to thank also Peter Clarkson for bringing reference [15] to our attention.

Author information

Authors and Affiliations

  1. Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile

    Marcel G. Clerc

  2. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Juan Diego Dávila & Michał Kowalczyk

  3. Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Panayotis Smyrnelis

  4. Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077, Göttingen, Germany

    Estefania Vidal-Henriquez

Authors
  1. Marcel G. Clerc
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  2. Juan Diego Dávila
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  3. Michał Kowalczyk
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  4. Panayotis Smyrnelis
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  5. Estefania Vidal-Henriquez
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Corresponding author

Correspondence to Panayotis Smyrnelis.

Additional information

Communicated by A. Malchiodi.

M. G. Clerc was partially supported by Fondecyt 1150507.

J. Davila was partially supported by Fondecyt 1170224, Fondo Basal CMM-Chile and Millennium Nucleus NC130017.

M. Kowalczyk was partially supported by Chilean research grants Fondecyt 1130126 and 1170164, Fondo Basal CMM-Chile.

P. Smyrnelis was partially supported by Fondo Basal CMM-Chile and Fondecyt postdoctoral Grant 3160055.

E. Vidal-Henriquez was partially supported by a Master fellowship CONICYT 221320023 and DPP of the University of Chile.

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Clerc, M.G., Dávila, J.D., Kowalczyk, M. et al. Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation. Calc. Var. 56, 93 (2017). https://doi.org/10.1007/s00526-017-1187-8

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