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Pattern formation in the models with coupling between order parameter and its gradient

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Abstract

A possibility of pattern formation in the models of the first order phase transitions with coupling between the order parameter and its gradient is discussed. We use the standard model of phase transitions extended to the higher derivatives of the order parameter that makes possible to describe the formation of various spatial distributions of the order parameter after phase transition. An example of the simple model with coupling between the order parameter and its gradient is considered. The proposed model is analogical to the mechanical nonlinear oscillator with the coordinate-dependent mass or velocity-dependent elastic module. The exact solution of this model is obtained that can be used to predict the order parameter distribution in the case of a spinodal decomposition.

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References

  1. L.D. Landau, E.M. Lifshitz,Statistical Physics (Nauka, Moscow, 1976)

  2. A.D. Linde,Elementary particle physics and inflationary cosmology (Horwood Academic, Switzerland, 1990)

  3. P. Chaikin, T.C. Lubensky,Principle of Condensed Matter Physics (Cambridge University Press, New York, 1998)

  4. P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd edn. (Clarendon Press, Oxford, 1993)

  5. J.S. Langer, Ann. Phys. 41, 108 (1967)

    Article  ADS  Google Scholar 

  6. P. Aviles, Y. Giga, Proc. Center Math Analysis Australian National University 12, 1 (1987)

    MathSciNet  Google Scholar 

  7. P. Aviles, Y. Giga, Proc. Roy. Soc. Edinburgh A 129, 1 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Mihailescu, arXiv:0710.5076v2 [cond-mat.soft] (2007)

  9. Z. Nussinov, I. Vekhter, A.V. Balatsky, Phys. Rev. B 79, 165122 (2009)

    Article  ADS  Google Scholar 

  10. M. Seul, D. Andelman, Science 267, 476 (1995)

    Article  ADS  Google Scholar 

  11. Z. Nusinov, arXiv:0105253v12 [cond-mat.statmech] (2004)

  12. S.B. Goryachev, Phys. Rev. Lett. 72, 1815 (1994)

    Article  Google Scholar 

  13. G. Gioia, M. Ortiz, Adv. Appl. Mech. 33, 119 (1997)

    Article  Google Scholar 

  14. M. Ortiz, G. Gioia, J. Mech. Phys. Solids 42, 531 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. N.M. Ercolani, R. Indik, A.C. Newell, T. Passot, J. Nonlin. Sci. 10, 223 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. S. Baldo, Ann. Inst. H. Poincaré 7, 67 (1990)

    MathSciNet  MATH  Google Scholar 

  17. R.A. Cowley, Phys. Rev. B 13, 4877 (1985)

    Article  ADS  Google Scholar 

  18. M. Cross, A. Newell, Physica D 10, 299 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. A. Newell, T. Passot, C. Bowman, N. Ercolani, R. Indik, Physica D 97, 185 (1996)

    Article  Google Scholar 

  20. E. Kamke, Differential Gleichunger, Losungs Methoden und Losungen (Verbesserte Auflage, Leipzig, 1959)

  21. Jingdong Bao I., Yizhong Zhuo, Xizhen Wu, Z. Phys. A 347, 217 (1994)

    Article  ADS  Google Scholar 

  22. Krishna Kumar, Prog. Theor. Phys. 84, 373 (1990)

    Article  ADS  Google Scholar 

  23. M. Yamamura, Prog. Theor. Phys. 64, 94 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. H. Kleinert, A. Chervyakov, arXiv:quant-ph/0206022v1 (2002)

  25. B. Delamotte, Phys. Rev. Lett. 70, 3361 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. I. Andrianov, J. Awrejcewicz, Int. J. Nonlin. Sci. Num. Simul. 1, 327 (2000)

    MathSciNet  MATH  Google Scholar 

  27. C.M. Bender, K.A. Milton, S.S. Pinsky, L.M. Simmons, J. Math. Phys. 30, 1447 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. A.H. Nayfeh,Introduction to Perturbation Techniques (Wiley, New York, 1981)

  29. Ji-Huan He, Int. J. Nonlinear Mech. 35, 37 (2000)

    Article  ADS  MATH  Google Scholar 

  30. Ji-Huan He, Int. J. Nonlinear Mech. 34, 699 (1999)

    Article  ADS  MATH  Google Scholar 

  31. Ji-Huan He, Phys. Rev. Lett. 90, 174301 (2003)

    Article  ADS  Google Scholar 

  32. W. Jin, R.V. Kohn, J. Nonlinear Sci. 10, 355 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Bogolyubov Institute for Theoretical Physics, NAS Ukraine, Metrologichna 14-b, 03680, Kyiv, Ukraine

    B. I. Lev & A. G. Zagorodny

  2. Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, 305-8568, Tsukuba, Japan

    B. I. Lev

Authors
  1. B. I. Lev
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  2. A. G. Zagorodny
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Correspondence to B. I. Lev.

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Lev, B.I., Zagorodny, A.G. Pattern formation in the models with coupling between order parameter and its gradient. Eur. Phys. J. B 86, 422 (2013). https://doi.org/10.1140/epjb/e2013-40674-1

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  • DOI: https://doi.org/10.1140/epjb/e2013-40674-1

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