Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Properties of a Periodic Ansatz for the Coarsening of Soliton-lattice

Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 73--86 | DOI:10.5890/DNC.2014.03.006

Simon Villain-Guillot

Laboratoire Onde et Matière d’Aquitaine, Université de Bordeaux, 351 cours de la Libéation 33405 Talence Cedex, France

Download Full Text PDF



Soliton lattices are periodic solutions of Ginzburg-Landau equation which can be useful tools to explore the coarsening process (or Ostwald ripening) which takes place during a Cahn-Hilliard dynamics.They can be used to identify the stationary solutions of the dynamics and how these intermedi- ate states are destroyed by fluctuations. The coarsening process drives the systems from a stationary solution to the next one which is of period double and of lower energy. Using another family of soliton lattices, this process can be described continuously via a phase field equation. We present here properties of these two families, including the Fourier series decomposition of the non symmetric soliton lattice which we use as building block of our ansatz.


  1. [1]  Izumitani, T. and Hashimoto, T. (1985), Slow spinodal decomposition in binary liquid mixtures of polymers, The Journal of Chemical Physics, 83, 3694 .
  2. [2]  Villain-Guillot, S. and Josserand, C. (2002), Non-linear growth of periodic interface, Physical Review E, 66, 036308.
  3. [3]  Langer, J.S. (1971), Theory of spinodal decomposition in alloys, Annals of Physics, 65, 53.
  4. [4]  Langer, J.S. (1992), An introduction to the kinetics of first-order phase transition, in: Solids Far From Equilibrium, 297-363, edited by C. Godr`eche, Cambridge University Press, Cambridge, England.
  5. [5]  Hillert, M. (1961), A Solid Solution Model for Inhomogeneous Systems, Acta Metallurgica , 9, 525.
  6. [6]  Cahn, J.W. and Hilliard, J.E. (1958), Free Energy of a Nonuniform System. I. Interfacial Free Energy, The Journal of Chemical Physics, 28, 258.
  7. [7]  Hohenberg, P.C. and Halperin, B.I. (1977), Theory of dynamical critical phenomena, Reviews of Modern Physics, 49, 435. See also M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993). For a review, see J.D. Guton, M. SanMiguel and P.S. Sahni, The dynamics of first order phase transitions, in:Phase Transition and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, London, 1983), Vol. 8, p. 267.
  8. [8]  Chevallard, C., Clerc, M., Coullet, P., and Gilli, J.M. (2000), Interface dynamics in Liquid crystals, The European Physical Journal E, 1, 179.
  9. [9]  Oyama, Y. (1939),Mixxing of solids, Bull. Inst. Phys. Chem. Res. Rep., 5, 600.
  10. [10]  Puri, S. and Hayakawa, H. (2001), Segregation of Granular Mixtures in a Rotating Drum, Advances in Complex Systems, 4(4), 469-0479.
  11. [11]  Scherer, M.A.,Melo, F., and Marder,M. (1999), Sand ripples in an oscillating annular sand–water cell, Physical Fluids, 11, 58.
  12. [12]  Stegner, A. andWesfreid, J.E. (1999), Dynamical evolution of sand ripples under water, Physical Review E, 60, R3487.
  13. [13]  Langer, J.S., Baron, M., and Miller, H.D. (1975), New computationalmethod in the theory of spinodal decomposition, Physical Review A, 11, 1417.
  14. [14]  Saxena, A. and Bishop, A.R. (1991), Multipolaron solutions of the Gross-Neveu field theory: Toda potential and doped polymers, Physical Review A, 44, R2251.
  15. [15]  Novik-Cohen, A. and Segel, L.A. (1984), Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10, 277.
  16. [16]  Abramowitz,M. and Stegun, I. (1965), Handbook of Mathematical Functions, Dover, New York.
  17. [17]  Arscott, F.M. (1981), Periodic Differential Equations, Pergamon, Oxford.
  18. [18]  Coullet, P., Goldstein, R.E., and Gunaratne, J.H. (1989), Parity-breaking transitions of modulated patterns in hydrodynamic systems, Physical Review Letters, 63, 1954.
  19. [19]  Politi, P. and Misbah, C. (2004), When does coarsening occur in the dynamics of one-dimensional fronts? Physical Review Letters, 92, 090601.