Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 3(19). C. 10-19. ISSN 2079-6641

Содержание выпуска

DOI: 10.18454/2079-6641-2017-19-3-10-19

MSC 35R01+76M60+17B66

CONSERVATION LAWS AND SYMMETRY ANALYSIS OF (1+1)-DIMENSIONAL SAWADA-KOTERA EQUATION

S. R. Hejazi, E. Lashkarian

Department of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.

E-mail: ra.hejazi@gmail.com, lashkarianelham@yahoo.com

The paper addresses an extended (1+1)-dimensional Sawada-Kotera (SK) equation. The Lie symmetry analysis leads to many plethora of solutions to the equation. The non-linear self-adjointness condition for the SK equation established and subsequently used to construct simplified independent conserved vectors. In particular, we also get conservation laws of the equation with the corresponding Lie symmetry.

Key words: Fluid mechanics, Lie symmetry, Partial differential equation, Shear stress, Optimal system, Partial differential equation; KdV equation, Lie symmetry; Conservation Laws.

References

  1. Alexandrova A. A., Ibragimov N. H., Imamutdinova K.V., Lukashchuk V. O., “Local and nonlocal conserved vectors for the nonlinear filtration equation”, Ufa Math J., . 9:4 (2012), 179–85.
  2. Kerishnan E.V., “On Sawada-Kotera equations”, II Nuovo Cimento B, 92:1, 23-26.
  3. Bluman G.W., Cheviakov A. F., Anco C., “Construction of Conservation Laws: How the Direct Method Generalizes Noether’s Theorem”, Proceeding of 4th Workshop «Group Analysis of Differential Equations & Integribility», 2009, 1–23.
  4. Bluman G.W., Cheviakov A. F., Anco C., Application of Symmetry Methods to Partial Differential Equations, Springer, New York, 2000.
  5. Bluman G.W., Cole J. D., “The general similarity solution of the heat equation”, J. Math. Mech., 18 (1969), 1025–1042.
  6. Fushchych W. I., Popovych R. O., “Symmetry reduction and exact solutions of the Navier–Stokes equations”, J. Nonlinear Math. Phys., 1:75–113 (1994), 156–188.
  7. Hejazi S. R., “Lie group analaysis, Hamiltonian equations and conservation laws of Born–Infeld equation”, Asian-European Journal of Mathematics, 7:3 (2014), 1450040.
  8. Hydon P. E., Stmmetry Method for Differential Equations, Cambridge University Press, UK, Cambridge, 2000.
  9. Ibragimov N. H., Transformation group applied to mathematical physics, Riedel, Dordrecht, 1985.
  10. Ibragimov N. H., Aksenov AV., Baikov V. A., Chugunov V. A. , Gazizov R. K. and
    Meshkov A. G., CRC handbook of Lie group analysis of differential equations. Applications in engineering and physical sciences. V. 2, CRC Press, Boca Raton, 1995.
  11. Ibragimov N. H., “Nonlinear self-adjointness in constructing conservation laws”, Arch ALGA, 7/8 (2010–2011), 1–99.
  12. Ibragimov N. H., Anderson R. L., “Lie theory of differential equations”, Lie group analysis of differential equations. Symmetries, exact solutions and conservation laws.. V. 1, CRC Press, Boca Raton, 1994, 7–14.
  13. Ibragimov N. H., N Arch ALGA 2010–2011, 7/8, 1–99.
  14. Li J. B., Wu J. H. and Zhu H. P., “Travelling Waves for an Integrable Higher Order KdV Type Wave Equation”, Int. J. Bifur Chaos Appl. Sci. Eng., 2006, 2235–2260.
  15. Nadjafikhah M., Hejazi S. R., “Symmetry analysis of cylindrical Laplace equation”, Balkan journal of geometry and applications, 2009.
  16. Olver P. J., Equivalence, Invariant and Symmetry, Cambridge, Cambridge University Press, 1995.
  17. Ovsiannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982.
  18. Zwillinger D., Handbook of Differential Equations, Academic Press, Boston, 1997, 132 pp.

 

References (GOST)

  1. Alexandrova A. A., Ibragimov N. H., Imamutdinova K.V., Lukashchuk V. O. Local and nonlocal conserved vectors for the nonlinear filtration equation // Ufa Math J. 2012. vol. 9. issue 4. pp. 179–85.
  2. Kerishnan E.V. On Sawada-Kotera equations // II Nuovo Cimento B. vol. 92. issue 1, pp. 23-26.
  3. Bluman G.W., Cheviakov A. F., Anco C. Construction of Conservation Laws: How the Direct Method Generalizes Noether’s Theorem // Proceeding of 4th Workshop «Group Analysis of Differential Equations & Integribility». 2009. pp. 1-23.
  4. Bluman G.W., Cheviakov A. F., Anco C. Application of Symmetry Methods to Partial Differential Equations. New York: Springer, 2000.
  5. Bluman G.W., Cole J. D. The general similarity solution of the heat equation // J. Math. Mech. 1969. vol. 18. pp. 1025–1042.
  6. Fushchych W. I., Popovych R. O. Symmetry reduction and exact solutions of the Navier–Stokes equations // J. Nonlinear Math. Phys. 1994. vol. 1. no. 75–113. pp. 156–188.
  7. Hejazi S. R. Lie group analaysis, Hamiltonian equations and conservation laws of Born–Infeld equation // Asian-European Journal of Mathematics. 2014. vol. 7. no. 3. 1450040.
  8. Hydon P. E. Stmmetry Method for Differential Equations. UK. Cambridge: Cambridge University Press, 2000
  9. Ibragimov N. H. Transformation group applied to mathematical physics. Riedel, Dordrecht, 1985.
  10. Ibragimov N. H., Aksenov AV., Baikov V. A., Chugunov V. A. , Gazizov R. K. and
    Meshkov A. G. CRC handbook of Lie group analysis of differential equations. In: Ibragimov NH, editor. Applications in engineering and physical sciences, vol. 2. Boca Raton: CRC Press; 1995.
  11. Ibragimov N. H. Nonlinear self-adjointness in constructing conservation laws. Arch ALGA 2010–2011. 7/8. pp. 1–99.
  12. Ibragimov N. H., Anderson R. L. Lie theory of differential equations. In: Ibragimov NH, editor. Lie group analysis of differential equations, vol 1, Symmetries, exact solutions and conservation laws. Boca Raton: CRC Press; 1994. pp. 7–14.
  13. Ibragimov N. H., N Arch ALGA 2010–2011. 7/8. pp. 1–99.
  14. Li J. B., Wu J. H. and Zhu H. P. Travelling Waves for an Integrable Higher Order KdV Type Wave Equation // Int. J. Bifur Chaos Appl. Sci. Eng. 2006. pp. 2235–2260.
  15. Nadjafikhah M., Hejazi S. R. Symmetry analysis of cylindrical Laplace equation // Balkan journal of geometry and applications. 2009.
  16. Olver P. J. Equivalence, Invariant and Symmetry. Cambridge: Cambridge University Press,1995.
  17. Olver P. J. Applications of Lie Groups to Differential equations. Second Edition. GTM, vol. 107. New York: Springer-Verlage, 1993.
  18. Ovsiannikov L.V. Group Analysis of Differential Equations. New York: Academic Press, 1982.
  19. Zwillinger D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997. p. 132.

Для цитирования: Hejazi S. R., Lashkarian E. Conservation laws and symmetry analysis of (1+1)-dimensional Sawada-Kotera equation // Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 3(19). C. 10-19. DOI: 10.18454/2079-6641-2017-19-3-10-19.
For citation: Hejazi S. R., Lashkarian E. Conservation laws and symmetry analysis of (1+1)-dimensional Sawada-Kotera equation, Vestnik KRAUNC. Fiz.-mat. nauki. 2017, 19: 3, 10-19. DOI: 10.18454/2079-6641-2017-19-3-10-19.

Поступила в редакцию / Original article submitted: 26.09.2017

    ХейязиResa Риза Сейед – кандидат физико-математических наук, доцент отдела математических наук Шахрудского технологического университета, г. Шахруд, провинция Семнан, Исламская Республика Иран.
     Hejazi Reza Seyed – Ph.D. (Differential Geometry), Assist. Profes  sor, Department of Mathematics Sciences, University of Shahrood, Shahrood, Semnan, Iran.

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Lashkarian      Лашкариан Эльхам – кандидат физико-математических наук, доцент отдела математических наук  Шахрудского технологического университета, г. Шахруд, провинция Семнан, Исламская Республика Иран.

     Lashkarian Elham – Ph.D. (Differential Geometry), Assist. Professor, Department of Mathematics  Sciences, University of Shahrood, Shahrood, Semnan, Iran.

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