HUANG Ding-jiang, ZHANG Hong-qing. Preliminary Group Classification of Quasi-Linear Third Order Evolution Equations[J]. Applied Mathematics and Mechanics, 2009, 30(3): 265-281.
Citation: HUANG Ding-jiang, ZHANG Hong-qing. Preliminary Group Classification of Quasi-Linear Third Order Evolution Equations[J]. Applied Mathematics and Mechanics, 2009, 30(3): 265-281.

Preliminary Group Classification of Quasi-Linear Third Order Evolution Equations

  • Received Date: 2008-06-24
  • Rev Recd Date: 2008-12-17
  • Publish Date: 2009-03-15
  • Group classification of quasilinear thir dorder evolution equations is performed by using the classical infinitesimal Lie method, the technique of equivalence transformations and the theory of classification of abstract low-dimensional Lie algebras. It is indicated that there are three equations admitting simple Lie algebras of dimension three. What's more, all the inequivalent equations admitting simple Lie algebra are nothing but them. Further more, it is also shown that there exist two, five, twenty-nine and twenty-six inequivalent third or der nonlinear evolution equations a dmitting one-, two, three, and fourdimensional solvable Lie algebras, respectively.
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  • [1]
    Bluman G,Anco S C.Symmetry and Integration Methods for Differential Equations[M].New York:Springer,2002.
    [2]
    Bluman G W,Kumei S.Symmetries and Differential Equations[M].New York:Springer,1989.
    [3]
    Fushchych W I,Shtelen W M,Serov N I.Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics[M].Dordrecht:Kluwer,1993.
    [4]
    Fushchych W I,Zhdanov R Z.Symmetries and Exact Solutions of Nonlinear Dirac Equations[M].Kyiv:Naukova Ukraina,1997.
    [5]
    Ibragimov N H.Transformation Groups Applied to Mathematical Physics[M].Dordrecht:D Reidel Publishing Co.,1985.
    [6]
    Ibragimov N H.Lie Group Analysis of Differential Equations—Symmetries,Exact Solutions and Conservation Laws[M].Vol 1.Boca Raton:CRC Press,1994.
    [7]
    Ibragimov N H.Elementary Lie Group Analysis and Ordinary Differential Equations[M].New York:Wiley,1999.
    [8]
    Olver P J.Application of Lie Groups to Differential Equations[M].New York:Springer-Verlag,1986.
    [9]
    Ovsiannikov L V.Group Analysis of Differential Equations[M].New York:Academic Press,1982.
    [10]
    Stephani H.Differential Equation:Their Solution Using Symmetries[M].Cambridge:Cambridge University Press,1994.
    [11]
    Lie S,Engel F.Theorie der Transformationsgruppen[M].3Bd.Leipzig:Teubner.1888,1890,1893.
    [12]
    Lie S.On integration of a class of Linear partial differential equations by means of definite integrals[A].In:Ibragimov N H Ed.CRC Handbook of Lie Group Analysis of Differential Equations[C]. Vol.2,Boca Raton:CRC Press, 1994,473-508.(Translation by Ibragimov N H of Arch for Math,Bd.VI,Heft 3,328-368,Kristiania 1881).
    [13]
    Gazeau J P,Winternitz P.Symmetries of variable coefficient Korteweg-de Vries equations[J].J Math Phys,1992,33(12):4087-4102. doi: 10.1063/1.529807
    [14]
    Güngr F,Lahno V I,Zhdanov R Z.Symmetry classification of KdV-type nonlinear evolution equations[J].J Math Phys,2004,45(6):2280-2313. doi: 10.1063/1.1737811
    [15]
    Basarab-Horwath P,Lahno V,Zhdanov R.The structure of Lie algebras and the classification problem for partial differential equations[J].Acta Applicandae Mathematicae,2001,69(1):43-94. doi: 10.1023/A:1012667617936
    [16]
    Bluman G,Temuerchaolu,Sahadevan R.Local and nonlocal symmetries for nonlinear telegraph equation[J].J Math Phys,2005,46(2):023505. doi: 10.1063/1.1841481
    [17]
    QU Chang-zheng.Allowed transformations and symmetry class of variable-coefficient Burgers equations[J].IMA J Appl Math,1995,54(3):203-225. doi: 10.1093/imamat/54.3.203
    [18]
    HUANG Ding-jiang,Ivanova N M.Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations[J].J Math Phys,2007,48(7):073507. doi: 10.1063/1.2747724
    [19]
    Zhdanov R Z,Lahno V I.Group classification of heat conductivity equations with a nonlinear source[J].J Phys A:Math Gen,1999,32:7405-7418. doi: 10.1088/0305-4470/32/42/312
    [20]
    Lahno V I,Zhdanov R Z.Group classification of nonlinear wave equations[J].J Math Phys,2005,46(5):053301. doi: 10.1063/1.1884886
    [21]
    Lahno V I,Zhdanov R Z,Magda O.Group classification and exact solutions of nonlinear wave equations[J].Acta Appl Math,2006,91(3):253-313. doi: 10.1007/s10440-006-9039-0
    [22]
    Zhdanov R Z,Lahno V I.Group classification of the general evolution equation:Local and quasilocal symmetries[J].Symmetry Integrability and Geometry:Methods and Applications,2005,1:009.
    [23]
    黄定江.非线性波、几何可积性与群分类[D].博士学位论文,大连:大连理工大学,2007.
    [24]
    Basarab-Horwath P,Gungor F,Lahno V.Symmetry classification of third-order nonlinear evolution equations[Z]. arXiv,2008,nlin.SI-0802.0367v1:1-73.
    [25]
    Gagnon L,Winternitz P.Symmetry classes of variable coefficient nonlinear Schrdinger equations[J].J Phys A:Math Gen,1993,26:7061-7076. doi: 10.1088/0305-4470/26/23/043
    [26]
    Gómez-Ullate D,Lafortune S,Winternitz P.Symmetries of discrete dynamical systems involving two species[J].J Math Phys,1999,40(6):2782-2804. doi: 10.1063/1.532728
    [27]
    Gungor F,Winternitz P.Generalized Kadomtsev-Petviashvili equation with an infinite dimensional symmetry algebra[J].J Math Anal Appl,2002,276:314-328. doi: 10.1016/S0022-247X(02)00445-6
    [28]
    Levi D,Winternitz P.Symmetries of discrete dynamical systems[J].J Math Phys,1996,37(11):5551-5576. doi: 10.1063/1.531722
    [29]
    Lafortune S,Tremblay S,Winternitz P.Symmetry classification of diatomic molecular chains[J].J Math Phys,2001,42(11):5341-5357. doi: 10.1063/1.1398583
    [30]
    Zhdanov R Z,Fushchych W I,Marko P V.New scale-invariant nonlinear differential equations for a complex scalar field[J].Physica D,1996,95(2):158-162. doi: 10.1016/0167-2789(96)00047-4
    [31]
    Zhdanov R,Roman O.On preliminary symmetry classification of nonlinear Schrdinger equation with some applications of Doebner-Goldin models[J].Rep Math Phys,2000,45:273-291. doi: 10.1016/S0034-4877(00)89037-0
    [32]
    Lie S.Gesammelte Abhandlungen[M].Vol 5.Leipzig:Teubner,1924.
    [33]
    Lie S.Gesammelte Abhandlungen[M].Vol 6.Leipzig:Teubner,1927.
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