Visn. Nac. Akad. Nauk Ukr. 2017. (9):33-40

O.O. Vaneeva
Institute of Mathematics of the National Academy of Sciences of Ukraine,Kyiv

According to the materials of scientific report at the meeting of the Presidium of NAS of Ukraine, July 5, 2017

The report is devoted to the problem of Lie symmetry classification for classes of nonlinear partial differential equations. Such symmetries allow one, in particular, to select equations of potential physical interest and to construct their exact solutions. For many classes of partial differential equations which are important for applications classical methods of group analysis do not result in exhaustive group classification. Such complicated group classification problems require new tools to be solved completely. Majority of the modern approaches are based on the usage of nondegenerate point transformations. Using the group classifications of variable coefficient generalized Kawahara equations and quasilinear reaction–diffusion equations as illustrative examples, we show the effectiveness of the recently developed approaches. These approaches include, in particular, the construction of the widest possible equivalence groups and the method of mapping between classes.
Keywords: Lie symmetry, group classification, equivalence group, method of mappings between classes, Kawahara equation, reaction-diffusion equation, exact solutions.

Language of article: ukrainian



1.     Dirac P.A.M. The evolution of the physicist’s picture of nature. Scientific American. 1963. 208(5): 47.

2.     Olver P.J. Applications of Lie groups to differential equations. Graduate Texts in Mathematics, 107. (New York: Springer-Verlag, 1993).

3.     Fushchich W.I., Nikitin A.G. Symmetries of equations of quantum mechanics. (New York: Allerton Press Inc., 1994).

4.     Ovsiannikov L.V. Group analysis of differential equations. (New York: Academic Press, 1982).

5.     Ibragimov N.H. (ed.). Handbook of Lie Group Analysis of Differential Equations. V. 1–3. (Boca Raton: CRC Press, 1994, 1995, 1996).

6.     Popovych R.O., Kunzinger M., Eshraghi H. Admissible transformations and normalized classes of nonlinear Schrödinger equations. Acta Appl. Math. 2010. 109(2): 315.

7.     Samoilenko A.M. (ed). Fushchych W.I. Selected Works. (Kyiv: Naukova Dumka, 2005).

8.     Popovych R.O. Classification of admissible transformations of differential equations. Collection of Works of Institute of Mathematics. 2006. 3(2): 239.

9.     Popovych R.O., Bihlo A. Symmetry preserving parameterization schemes. J. Math. Phys. 2012. 53(7): 073102.

10. Vaneeva O.O., Popovych R.O., Sophocleous C. Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source. Acta Appl. Math. 2009. 106(1): 1.

11. Ivanova N.M., Popovych R.O., Sophocleous C. Group analysis of variable coefficient diffusion-convection equations. II. Contractions and exact solutions. arXiv: 0710.3049.

12. Vaneeva O.O., Popovych R.O, Sophocleous C. Extended group analysis of variable coefficient reaction-diffusion equations with exponential nonlinearities. J. Math. Anal. Appl. 2012. 396(1): 225.

13. Nikitin A.G., Popovych R.O. Group classification of nonlinear Schrödinger equations. Ukr. Math. J. 2001. 53(8):1255.

14. Popovych R.O., Ivanova N.M. New results on group classification of nonlinear diffusion-convection equations, J. Phys. A. 2004. 37(30): 7547.

15. Vaneeva O. Lie symmetries and exact solutions of variable coefficient mKdV equations: An equivalence based approach. Commun. Nonlinear Sci. Numer. Simulat. 2012. 17(2): 611.

16. Kuriksha O., Pošta S., Vaneeva O. Group classification of variable coefficient generalized Kawahara equations. J. Phys. A: Math. Theor. 2014. 47(4): 045201.

17. Kawahara T. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 1972. 33: 260.

18. Kamin S., Rosenau P. Nonlinear thermal evolution in an inhomogeneous medium. J. Math. Phys. 1982. 23(7): 1385.

19. Vaneeva O., Kuriksha O., Sophocleous C. Enhanced group classification of Gardner equations with time-dependent coefficients. Commun. Nonlinear Sci. Numer. Simulat. 2015. 22(1-3): 1243.

20. Vaneeva O.O., Popovych R.O., Sophocleous C. Group analysis of Benjamin–Bona–Mahony equations with time dependent coefficients. J. Phys.: Conf. Ser. 2015. 621(1): 012016.

21. Vaneeva O., Pošta S., Sophocleous C. Enhanced group classification of Benjamin–Bona–Mahony–Burgers equations. Appl. Math. Lett. 2017. 65: 19.

22. Vaneeva O.O., Sophocleous C., Leach P.G.L. Lie symmetries of generalized Burgers equations: application to boundary-value problems. J. Eng. Math. 2015. 91(1): 165.

23. Vaneeva O., Karadzhov Yu., Sophocleous C. Group analysis of a class of nonlinear Kolmogorov equations. In: Lie theory and its applications in physics. Springer Proc. Math. Stat. (Singapore: Springer, 2016). P. 349–360.

24. Vaneeva O.O., Popovych R.O., Sophocleous C. Equivalence transformations in the study of integrability. Physica scripta. 2014. 89(3): 038003.