Skip to main content
SpringerLink
Log in

Bifurcations of Spatially Inhomogeneous Solutions in a Modified Version of the Kuramoto–Sivashinsky Equation

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

A periodic boundary-value problem for an equation with a deviating spatial argument is considered. Using the Poincaré–Dulac method of normal forms, the method of integral manifolds, and asymptotic formulas, we examine a number of bifurcation problems of codimension 1 and 2. For homogeneous equilibrium states, we analyze possibilities of implementing critical cases of various types. The problem on the stability of homogeneous equilibrium states is studied and asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Bachurin, A. S. Rudy, and V. K. Smirnov, “Nanoscale model of surface erosion by ion bombardment,” Radiat. Eff. Defects Solids., 161, No. 6, 319–329 (2006).

    Article  Google Scholar 

  2. R. M. Bradley and J. M. F. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Techn. A., 6, No. 4, 2390–2395 (1988).

    Article  Google Scholar 

  3. A. Yu. Kolesov and A. N. Kulikov, Invariant Tori of Nonlinear Evolutionary Equations [in Russian], Yaroslavl Univ., Yaroslavl (2005).

    Google Scholar 

  4. A. Yu. Kolesov, A. N. Kulikov, and N. Kh. Rozov, “Invariant tori of one class of point mappings: conservation of invariant tori under perturbations,” Differ. Uravn., 39, No. 6, 738–753 (2003).

    MathSciNet  Google Scholar 

  5. A. Yu. Kolesov and N. Kh. Rozov, Invariant Tori of Nonlinear Wave Equations [in Russian], Fizmatlit, Moscow (2004).

    Google Scholar 

  6. A. M. Kovaleva, A. N. Kulikov, and D. A. Kulikov, “Stability and bifurcations of wave-type solutions for a functional differential equation,” Izv. Inst. Mat. Inform. Udmurt. Univ., 2, No. 46, 60–68 (2015).

    Google Scholar 

  7. A. N. Kulikov, “On smooth invariant manifolds of a semigroup of nonlinear operators in Banach spaces,” in: Research on Stability and Oscillation Theory [in Russian], Yaroslavl Univ., Yaroslavl (1976), pp. 114–129.

  8. A. N. Kulikov and D. A. Kulikov, “Formation of wave-like nanostructures on the surface of flat substrates during ion bombardment,” Zh. Vychisl. Mat. Mat. Fiz., 52, No. 5, 930–945 (2012).

    MathSciNet  Google Scholar 

  9. A. N. Kulikov and D. A. Kulikov, “Nonlocal model of relief formation under the influence of ion flow. Inhomogeneous nanostructures,” Izv. Inst. Mat. Inform. Udmurt. Univ., 28, No. 3, 33–50 (2015).

    MathSciNet  Google Scholar 

  10. A. N. Kulikov and D. A. Kulikov, “Spatially inhomogeneous solutions for a modified Kuramoto-Sivashinsky equation,” J. Math. Sci., 219, No. 2, 173–183 (2016).

    Article  MathSciNet  Google Scholar 

  11. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin (1984).

    Book  Google Scholar 

  12. J. E. Marsden and M. McCracken, The Hopf bifurcation and Its Applications, Springer-Verlag, New York–Heidelberg–Berlin (1980).

  13. A. S. Rudyi and V. I. Bachurin, “Spatially nonlocal model of model of surface erosion by ion bombardment,” Izv. Ross. Akad. Nauk. Ser. Fiz., 72, No. 5, 624–629 (2008).

    Google Scholar 

  14. A. S. Rudyi, A. N. Kulikov, and A. V. Metlitskaya, “Modeling of nanostructure formation processes during sputtering by ion bombardment,” Mikroelektronika, 40, No. 2, 109–118 (2011).

    Google Scholar 

  15. P. Sigmund, “Theory of sputtering. I. Sputtering yield of amorphous and polycrystalline targets,” Phys. Rev., 184, No. 2, 383–416 (1969).

    Article  Google Scholar 

  16. P. Sigmund, “A mechanism of surface micro-rougening by ion bombardment,” J. Math. Sci., 8, 1545 (1973).

    Article  Google Scholar 

  17. P. E. Sobolevsky, “On parabolic equations in Banach spaces,” Tr. Mosk. Mat. Obshch., 10, 297–350 (1967).

    Google Scholar 

  18. G. I. Syvashinsky, “Weak Turbulence in Periodic Flow,” Phys. D. Nonlin. Phenom., 17, No. 2, 243–255 (1985).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yaroslavl State University, Yaroslavl, Russia

    A. M. Kovaleva

Authors
  1. A. M. Kovaleva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Kovaleva.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 185, Proceedings of the All-Russian Scientific Conference “Differential Equations and Their Applications” Dedicated to the 85th Anniversary of Professor M. T. Terekhin. Ryazan State University named for S. A. Yesenin, Ryazan, May 17-18, 2019. Part 1, 2020.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kovaleva, A.M. Bifurcations of Spatially Inhomogeneous Solutions in a Modified Version of the Kuramoto–Sivashinsky Equation. J Math Sci 281, 398–411 (2024). https://doi.org/10.1007/s10958-024-07114-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-024-07114-z

Keywords and phrases

AMS Subject Classification

Search

Navigation