Ukrainian Mathematical Journal

, Volume 58, Issue 8, pp 1275–1297 | Cite as

Classification of linear representations of the Galilei, Poincaré, and conformal algebras in the case of a two-dimensional vector field and their applications

  • M. I. Serov
  • T. O. Zhadan
  • L. M. Blazhko


We present the classification of linear representations of the Galilei, Poincaré, and conformal algebras nonequivalent under linear transformations in the case of a two-dimensional vector field. The obtained results are applied to the investigation of the symmetry properties of systems of nonlinear parabolic and hyperbolic equations.


Linear Representation Arbitrary Constant Symmetry Property Equivalence Transformation Projective Transformation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Fushchych and R. Cherniha, “Galilei-invariant systems of nonlinear systems of evolution equations,” J. Phys. A: Math. Gen., 5569–5579 (1995).Google Scholar
  2. 2.
    G. Rideau and P. Winternitz, “Evolution equations invariant under two-dimensional space-time Schoding groups,” J. Math. Phys., 558–569 (1993).Google Scholar
  3. 3.
    R. M. Cherniha and J. R. King, “Lie symmetries of nonlinear multidimensional reaction diffusion systems: II,” J. Phys. A: Math. Gen., 405–425 (2003).Google Scholar
  4. 4.
    R. Cherniha and M. Serov, “Nonlinear systems of the Burgers-type equations: Lie and Q-conditional symmetries, ansatze, and solutions,” J. Math. Anal. Appl., 305–328 (2001).Google Scholar
  5. 5.
    A. G. Nikitin and R. Wiltshire, “Systems of reaction diffusion equations and their symmetry properties,” J. Math. Phys., 42, No. 4, 1666–1688 (2001).CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    V. I. Lahno, S. V. Spichak, and V. Stohnii, Symmetry Analysis of Evolutionary Equations [in Ukrainian], 45, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2002).Google Scholar
  7. 7.
    I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Group properties and exact solutions of the equations of nonlinear filtration,” in: Numerical Methods for the Solution of Problems of Filtration of Multiphase Incompressible Liquid [in Russian], Novosibirsk (1987), pp. 24–27.Google Scholar
  8. 8.
    I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Nonlocal symmetries. Heuristic Approach,” in: Itogi VINITI, Contemporary Problems of Mathematics, Recent Advances [in Russian], Vol. 34, VINITI, Moscow (1989), pp. 3–83.Google Scholar
  9. 9.
    A. V. Hleba, Symmetry Properties and Exact Solutions of Nonlinear Galilei-Invariant Equations [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Kyiv (2003).Google Scholar
  10. 10.
    W. Fushchych, W. Shtelen, and M. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht (1993).Google Scholar
  11. 11.
    L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).zbMATHGoogle Scholar
  12. 12.
    P. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin (1993).zbMATHGoogle Scholar
  13. 13.
    M. I. Serov and R. M. Cherniha, “Lie symmetries, Q-conditional symmetries, and exact solutions of a system of nonlinear diffusion-convection equations,” Ukr. Mat. Zh., 55, No. 10, 1340–1355 (2003).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • M. I. Serov
    • 1
  • T. O. Zhadan
    • 1
  • L. M. Blazhko
    • 1
  1. 1.Poltava State Technical UniversityPoltava

Personalised recommendations