Advertisement

Buffer phenomenon in the spatially one-dimensional Swift-Hohenberg equation

  • A. Yu. Kolesov
  • E. F. Mishchenko
  • N. Kh. RozovEmail author
Article

Abstract

We consider a boundary value problem for the spatially one-dimensional Swift-Hohenberg equation with zero Neumann boundary conditions at the endpoints of a finite interval. We establish that as the length l of the interval increases while the supercriticality ɛ is fixed and sufficiently small, the number of coexisting stable equilibrium states in this problem indefinitely increases; i.e., the well-known buffer phenomenon is observed. A similar result is obtained in the 2l-periodic case.

Keywords

Equilibrium State STEKLOV Institute Solvability Condition Dissipative Structure Instability Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Swift and P. C. Hohenberg, “Hydrodynamic Fluctuations at the Convective Instability,” Phys. Rev. A 15(1), 319–328 (1977).CrossRefGoogle Scholar
  2. 2.
    H. Haken, Advanced Synergetics (Springer, Berlin, 1983).zbMATHGoogle Scholar
  3. 3.
    A. V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics (World Sci., Singapore, 1998; Editorial URSS, Moscow, 1999).zbMATHGoogle Scholar
  4. 4.
    M. Tlidi, M. Georgiou, and P. Mandel, “Transverse Patterns in Nascent Optical Bistability,” Phys. Rev. A 48(6), 4605–4609 (1993).CrossRefGoogle Scholar
  5. 5.
    J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg Equation for Lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).CrossRefGoogle Scholar
  6. 6.
    N. E. Kulagin, L. M. Lerman, and T. G. Shmakova, “On Radial Solutions of the Swift-Hohenberg Equation,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 261, 188–209 (2008) [Proc. Steklov Inst. Math. 261, 183–203 (2008)].MathSciNetGoogle Scholar
  7. 7.
    E. F. Mishchenko, V. A. Sadovnichii, A. Yu. Kolesov, and N. Kh. Rozov, Autowave Processes in Nonlinear Media with Diffusion (Fizmatlit, Moscow, 2005) [in Russian].Google Scholar
  8. 8.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations (Nauka, Moscow, 1974) [in Russian].zbMATHGoogle Scholar
  9. 9.
    P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley-Intersci., London, 1971; Mir, Moscow, 1973).zbMATHGoogle Scholar
  10. 10.
    Yu. A. Mitropol’skii and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics (Nauka, Moscow, 1973) [in Russian].Google Scholar
  11. 11.
    V. I. Arnold, Supplementary Chapters to the Theory of Ordinary Differential Equations (Nauka, Moscow, 1978) [in Russian].Google Scholar
  12. 12.
    A. Yu. Kolesov and N. Kh. Rozov, Invariant Tori of Nonlinear Wave Equations (Fizmatlit, Moscow, 2004) [in Russian].Google Scholar
  13. 13.
    P. Collet and J.-P. Eckmann, Instabilities and Fronts in Extended Systems (Princeton Univ. Press, Princeton, NJ, 1990).zbMATHGoogle Scholar
  14. 14.
    W. Eckhaus, Studies in Non-linear Stability Theory (Springer, Berlin, 1965).zbMATHGoogle Scholar
  15. 15.
    Y. Pomeau and S. Zaleski, “Wavelength Selection in One-Dimensional Cellular Structures,” J. Phys. 42(4), 515–528 (1981).MathSciNetGoogle Scholar
  16. 16.
    S. Kogelman and R. C. DiPrima, “Stability of Spatially Periodic Supercritical Flows in Hydrodynamics,” Phys. Fluids 13(1), 1–11 (1970).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Y. Pomeau and P. Manneville, “Wavelength Selection in Cellular Flows,” Phys. Lett. A 75(4), 296–298 (1980).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • A. Yu. Kolesov
    • 1
  • E. F. Mishchenko
    • 2
  • N. Kh. Rozov
    • 3
    Email author
  1. 1.Yaroslavl State UniversityYaroslavlRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  3. 3.Moscow State UniversityMoscowRussia

Personalised recommendations