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

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.

This is a preview of subscription content, log in to check access.

Fig. 1

Notes

  1. 1.

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

  2. 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.

References

  1. 1.

    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  2. 2.

    Aftalion, A., Alama, S., Bronsard, L.: Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein condensate. Arch. Ration. Mech. Anal. 178(2), 247–286 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Aftalion, A., Blanc, X.: Existence of vortex-free solutions in the Painlevé boundary layer of a Bose-Einstein condensate. J. Math. Pures Appl. 83(6), 765–801 (2004)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Aftalion, A., Blanc, X., Dalibard, J.: Vortex patterns in a fast rotating Bose-Einstein condensate. Phys. Rev. A 71, 023611 (2005)

    Article  Google Scholar 

  5. 5.

    Aftalion, A., Jerrard, R.L., Royo-Letelier, J.: Non-existence of vortices in the small density region of a condensate. J. Funct. Anal. 260(8), 2387–2406 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Alikakos, N.D., Fife, P.C., Fusco, G., Sourdis, C.: Singular perturbation problem arising from the anisotropy of crystalline grain boundaries. J. Dyn. Differ. Equ. 19, 935–949 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Alikakos, N.D., Bates, P.W., Cahn, J.W., Fife, P.C., Fusco, G., Tanoglu, G.B.: Analysis of a corner layer problem in anisotropic interfaces. Discret. Contin. Dyn. Syst. Ser. B 6(2), 237–255 (2006)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Antonopoulos, P., Smyrnelis, P.: On minimizers of the Hamiltonian system \(u^{\prime \prime }=\nabla W(u)\), and on the existence of heteroclinic, homoclinic and periodic orbits. Indiana Univ. Math. J 65(5), 1503–1524 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Barboza, R., Bortolozzo, U., Clerc, M.G., Residori, S., Vidal-Henriquez, E.: Optical vortex induction via light-matter interaction in liquid-crystal media. Adv. Opt. Photonics 7, 635–683 (2015)

    Article  Google Scholar 

  10. 10.

    Barboza, R., Bortolozzo, U., Assanto, G., Vidal-Henriquez, E., Clerc, M.G., Residori, S.: Harnessing optical vortex lattices in nematic liquid crystals. Phys. Rev. Lett. 111, 093902 (2013)

    Article  Google Scholar 

  11. 11.

    Barboza, R., Bortolozzo, U., Assanto, G., Vidal-Henriquez, E., Clerc, M.G., Residori, S.: Vortex induction via anisotropy stabilized light-matter interaction. Phys. Rev. Lett. 109, 143901 (2012)

    Article  Google Scholar 

  12. 12.

    Barboza, R., Bortolozzo, U., Assanto, G., Vidal-Henriquez, E., Clerc, M.G., Residori, S.: Light-matter interaction induces a single positive vortex with swirling arms. Philos. Trans. R. Soc. A 372, 20140019 (2014)

    Article  Google Scholar 

  13. 13.

    Barboza, R., Bortolozzo, U., Davila, J.D., Kowalczyk, M., Residori, S., Vidal, E.: Henriquez, light-matter interaction induces a shadow vortex. Phys. Rev. E 90, 05201 (2016)

    Google Scholar 

  14. 14.

    Jonathan Chapman, S.: Superheating field of type II superconductors. SIAM J. Appl. Math. 55(5), 1233–1258 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Claeys, T., Kuijlaars, A.B.J., Vanlessen, M.: Multi-critical unitary random matrix ensembles and the general Painleve II equation. Ann. Math. 168(2), 601–641 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Clarkson, P.A.: Asymptotics of the second Painlevé equation. In: Diego, D., Robert, M., (eds) Special Functions and Orthogonal Polynomials, Contemp. Math., vol. 471, pp. 69–83. American Mathematical Society, Providence (2008)

  17. 17.

    Clerc, M.G., Vidal-Henriquez, E., Davila, J.D., Kowalczyk, M.: Symmetry breaking of nematic umbilical defects through an amplitude equation. Phys. Rev. E 90, 012507 (2014)

    Article  Google Scholar 

  18. 18.

    de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Oxford Science Publications, Clarendon Press, Wotton-under-Edg (1993)

    Google Scholar 

  19. 19.

    Deift, P.: Universality for mathematical and physical systems. International Congress of Mathematicians. Vol. I, p.125–152, Eur. Math. Soc., Zürich, (2007)

  20. 20.

    Flaschka, H., Newell, A.C.: Monodromy- and spectrum-preserving deformations I. Commun. Math. Phys. 76(1), 65–116 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Frisch, T.: Spiral waves in nematic and cholesteric liquid crystals. Phys. D 84, 601–614 (1995)

    Article  MATH  Google Scholar 

  22. 22.

    Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Ration. Mech. Anal. 73(1), 31–51 (1980)

    Article  MATH  Google Scholar 

  23. 23.

    Helffer, B., Weissrel, F.B.: On a family of solutions of the second Painlevé equation related to superconductivity. Eur. J. Appl. Math. 9(3), 223–243 (1998)

    Article  Google Scholar 

  24. 24.

    Ignat, R., Millot, V.: The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate. J. Funct. Anal. 233(1), 260–306 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Ignat, R., Millot, V.: Energy expansion and vortex location for a two dimensional rotating Bose-Einstein condensate. Rev. Math. Phys. 18(2), 119–162 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Its, A.R., Kapaev, A.A.: Quasi-linear stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16(1), 363 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Kapaev, A.A.: Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlevé equation. eprint arXiv:nlin/0411009 (2004)

  28. 28.

    Kapaev, A.A., Novokshenov, V.Y., Fokas, A.S., Its, A.R.: Painlevé transcendents: the Riemann-Hilbert approach. In: Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2006)

  29. 29.

    Karali, G., Sourdis, C.: The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit. Arch. Ration. Mech. Anal. 217(2), 439–523 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Karali, G., Sourdis, C.: Radial and bifurcation non-radial solutions of a singular perturbation problem in the case of exchange of stabilities. Ann. Inst. Heri Poincaré Anal. Non Linéaire 29, 131–170 (2012)

    Article  MATH  Google Scholar 

  31. 31.

    Karali, G., Sourdis, C.: Resonance phenomena in a singular perturbation problem in the case of exchange of stabilities. Commun. Partial Differ. Equ. 37, 1620–1667 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Kudryashov, N.A.: The second Painlevé equation as a model for the electric field in a semiconductor. Phys. Lett. A 233(4), 397–400 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Lassoued, L., Mironescu, P.: Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math 77, 1–26 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Palamides, P.K.: Generalized Painlevé equation and superconductivity. asymptotic behavior of unbounded solutions. Math. Comput. Model. 38(1), 177–189 (2003)

    Article  MATH  Google Scholar 

  35. 35.

    Residori, S.: Patterns, fronts and structures in a liquid-crystal-light-valve with optical feedback. Phys. Rep. 416, 201 (2005)

    Article  Google Scholar 

  36. 36.

    Senthilkumaran, M., Pandiaraja, D., Mayil Vaganan, B.: Exact and explicit solutions of Euler-Painlevé equations through generalized Cole-Hopf transformations. Appl. Math. Comput. 217(7), 3412–3416 (2010)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Sourdis, C., Fife, P.C.: Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces. Adv. Differ. Equ. 12, 623–668 (2007)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Troy, W: The role of Painlevé II in predicting new liquid crystal self-assembly mechanism, Preprint (2016)

Download references

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

Affiliations

Authors

Corresponding author

Correspondence to Panayotis Smyrnelis.

Additional information

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.

Communicated by A. Malchiodi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Mathematics Subject Classification

  • 35J20
  • 35J61
  • 35Q56
  • 35Q60