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Zh. Vychisl. Mat. Mat. Fiz., 2012, Volume 52, Number 5, Pages 930–945 (Mi zvmmf9720)  

This article is cited in 19 scientific papers (total in 19 papers)

Formation of wavy nanostructures on the surface of flat substrates by ion bombardment

A. N. Kulikov, D. A. Kulikov

Faculty of Mathematics, Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000 Russia

Abstract: A popular mathematical model for the formation of an inhomogeneous topography on the surface of a plate (flat substrate) during ion bombardment was considered. The model is described by the Bradley–Harper equation, which is frequently referred to as the generalized Kuramoto–Sivashinsky equation. It was shown that inhomogeneous topography (nanostructures in the modern terminology) can arise when the stability of the plane incident wavefront changes. The problem was solved using the theory of dynamical systems with an infinite-dimensional phase space, in conjunction with the integral manifold method and Poincaré–Dulac normal forms. A normal form was constructed using a modified Krylov–Bogolyubov algorithm that applies to nonlinear evolutionary boundary value problems. As a result, asymptotic formulas for solutions of the given nonlinear boundary value problem were derived.

Key words: nonlinear boundary value problem for the Bradley–Harper equation, stability of the solution, local bifurcations, quasi-normal form.

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English version:
Computational Mathematics and Mathematical Physics, 2012, 52:4, 800–814

Bibliographic databases:

UDC: 519.634
Received: 26.09.2011

Citation: A. N. Kulikov, D. A. Kulikov, “Formation of wavy nanostructures on the surface of flat substrates by ion bombardment”, Zh. Vychisl. Mat. Mat. Fiz., 52:5 (2012), 930–945; Comput. Math. Math. Phys., 52:4 (2012), 800–814

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    Erratum

    This publication is cited in the following articles:
    1. D. A. Kulikov, “Mekhanizm formirovaniya volnovykh dissipativnykh struktur v odnoi iz zadach nanotekhnologii”, Vestnik RAEN, 13:4 (2013), 23–31  mathscinet  elib
    2. A. M. Kovaleva, D. A. Kulikov, “Odnomodovye i dvukhmodovye neodnorodnye dissipativnye struktury v nelokalnoi modeli erozii”, Model. i analiz inform. sistem, 22:5 (2015), 665–681  mathnet  crossref  mathscinet  elib
    3. A. M. Kovaleva, A. N. Kulikov, D. A. Kulikov, “Ustoichivost i bifurkatsii volnoobraznykh reshenii dlya odnogo funktsionalno-differentsialnogo uravneniya”, Izv. IMI UdGU, 2015, no. 2(46), 60–68  mathnet  elib
    4. A. N. Kulikov, D. A. Kulikov, “Nelokalnaya model formirovaniya relefa pod vozdeistviem potoka ionov. Neodnorodnye nanostruktury”, Matem. modelirovanie, 28:3 (2016), 33–50  mathnet  elib
    5. A. I. Zemlyanukhin, A. V. Bochkarev, “Tochnye uedinenno-volnovye resheniya uravnenii Byurgersa–Khaksli i Bredli–Kharpera”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:1 (2017), 62–70  mathnet  crossref  elib
    6. A. V. Sekatskaya, “Bifurkatsii prostranstvenno neodnorodnykh reshenii v odnoi kraevoi zadache dlya obobschennogo uravneniya Kuramoto–Sivashinskogo”, Model. i analiz inform. sistem, 24:5 (2017), 615–628  mathnet  crossref  elib
    7. A. N. Kulikov, D. A. Kulikov, “Local bifurcations in the periodic boundary value problem for the generalized Kuramoto–Sivashinsky equation”, Autom. Remote Control, 78:11 (2017), 1955–1966  mathnet  crossref  isi  elib
    8. A. N. Kulikov, D. A. Kulikov, “Uravnenie Kuramoto–Sivashinskogo. Lokalnyi attraktor, zapolnennyi neustoichivymi periodicheskimi resheniyami”, Model. i analiz inform. sistem, 25:1 (2018), 92–101  mathnet  crossref  elib
    9. D. A. Kulikov, A. V. Sekatskaya, “O vliyanii geometricheskikh kharakteristik oblasti na strukturu nanorelefa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:3 (2018), 293–304  mathnet  crossref  elib
    10. Kulikov A.N., Kulikov D.A., “The Kuramoto-Sivashinsky Equation. a Local Attractor Filled With Unstable Periodic Solutions”, Autom. Control Comp. Sci., 52:7 (2018), 708–713  crossref  mathscinet  isi  scopus
    11. A. N. Kulikov, A. V. Sekatskaya, “Lokalnye attraktory v odnoi kraevoi zadache dlya uravneniya Kuramoto—Sivashinskogo”, Materialy mezhdunarodnoi konferentsii «Geometricheskie metody v teorii upravleniya i matematicheskoi fizike: differentsialnye uravneniya, integriruemost, kachestvennaya teoriya» Ryazan, 15–18 sentyabrya 2016 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 148, VINITI RAN, M., 2018, 58–65  mathnet  mathscinet
    12. A. M. Kovaleva, D. A. Kulikov, “Bifurkatsii prostranstvenno neodnorodnykh reshenii v dvukh versiyakh nelokalnogo uravneniya erozii”, Materialy mezhdunarodnoi konferentsii «Geometricheskie metody v teorii upravleniya i matematicheskoi fizike: differentsialnye uravneniya, integriruemost, kachestvennaya teoriya» Ryazan, 15–18 sentyabrya 2016 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 148, VINITI RAN, M., 2018, 66–74  mathnet  mathscinet
    13. A. V. Sekatskaya, “Issledovanie sostoyanii ravnovesiya vtorogo roda uravneniya Kuramoto – Sivashinskogo s odnorodnymi usloviyami Neimana”, Kompyuternye issledovaniya i modelirovanie, 11:1 (2019), 59–69  mathnet  crossref
    14. A. V. Sekatskaya, “Sostoyaniya ravnovesiya vtorogo roda uravneniya Kuramoto—Sivashinskogo s odnorodnymi kraevymi usloviyami Neimana”, Materialy mezhdunarodnoi konferentsii “Geometricheskie metody v teorii upravleniya i matematicheskoi fizike”, posvyaschennoi 70-letiyu S.L. Atanasyana, 70-letiyu I.S. Krasilschika, 70-letiyu A.V. Samokhina, 80-letiyu V.T. Fomenko. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 25–28 sentyabrya 2018 g. Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 168, VINITI RAN, M., 2019, 80–90  mathnet  crossref
    15. A. M. Kovaleva, “Bifurkatsii prostranstvenno neodnorodnykh reshenii v modifitsirovannom variante uravneniya Kuramoto—Sivashinskogo”, Materialy Vserossiiskoi nauchnoi konferentsii «Differentsialnye uravneniya i ikh prilozheniya», posvyaschennoi 85-letiyu professora M. T. Terekhina. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 17–18 maya 2019 g. Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 185, VINITI RAN, M., 2020, 58–71  mathnet  crossref
    16. A. V. Sekatskaya, “O kharaktere lokalnykh bifurkatsii uravneniya Kuramoto—Sivashinskogo v razlichnykh oblastyakh”, Materialy Vserossiiskoi nauchnoi konferentsii «Differentsialnye uravneniya i ikh prilozheniya», posvyaschennoi 85-letiyu professora M. T. Terekhina. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 17–18 maya 2019 g. Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 185, VINITI RAN, M., 2020, 72–78  mathnet  crossref
    17. D. A. Kulikov, “O lokalnykh bifurkatsiyakh prostranstvenno neodnorodnykh reshenii v odnom funktsionalno-differentsialnom uravnenii”, Materialy Vserossiiskoi nauchnoi konferentsii «Differentsialnye uravneniya i ikh prilozheniya», posvyaschennoi 85-letiyu professora M. T. Terekhina. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 17–18 maya 2019 g. Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 186, VINITI RAN, M., 2020, 67–73  mathnet  crossref
    18. A. N. Kulikov, D. A. Kulikov, “Cahn–Hilliard equation with two spatial variables. Pattern formation”, Theoret. and Math. Phys., 207:3 (2021), 782–798  mathnet  crossref  crossref  isi  elib
    19. A. N. Kulikov, D. A. Kulikov, “Attraktor obobschennogo uravneniya Kana—Khilliarda, vse resheniya na kotorom neustoichivy”, Materialy mezhdunarodnoi konferentsii po matematicheskomu modelirovaniyu v prikladnykh naukakh “International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19”. Belgorod, 20–24 avgusta 2019 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 195, VINITI RAN, M., 2021, 57–67  mathnet  crossref
  • Æóðíàë âû÷èñëèòåëüíîé ìàòåìàòèêè è ìàòåìàòè÷åñêîé ôèçèêè Computational Mathematics and Mathematical Physics
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