Abstract
We study the Krichever-Novikov equation from the standpoint of the theory of symmetry reductions in partial differential equations. We obtain a Lie group classification. Moreover, we obtain some exact solutions, and we apply the nonclassical method.
Similar content being viewed by others
References
S. I. Svinolupov and V. V. Sokolov, Funct. Anal. Appl., 16, 317–319 (1982).
R. Hernández Heredero, V. V. Sokolov, and S. I. Svinolupov, Phys. D, 87, 32–36 (1995).
V. V. Sokolov, Russ. Math. Surveys, 43, No. 5, 165–204 (1988).
V. E. Adler, Internat. Math. Res. Notices, 1, 1–4 (1998); arXiv:solv-int/9707015v1 (1997).
I. M. Krichever and S. P. Novikov, Russ. Math. Surveys, 35, No. 6, 53–79 (1980).
S. Igonin and R. Martini, J. Phys. A, 35, 9801–9810 (2002); arXiv:nlin/0208006v2 (2002).
N. Euler and M. Euler, J. Nonlin. Math. Phys., 16(Suppl. 1), 93–106 (2009).
S. I. Svinolupov, V. V. Sokolov, and R. I. Yamilov, Sov. Math. Dokl., 28, 165–168 (1983); arXiv:nlin/0110027v1 (2001).
F. W. Nijhoff, Phys. Lett. A, 297, 49–58 (2002).
N. Kh. Ibragimov, Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983); English transl.: Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht (1985).
P. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts Math., Vol. 107), Springer, New York (1993).
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978); English transl., Acad. Press, New York (1982).
G. W. Bluman and J. D. Cole, J. Math. Mech., 18, 1025–1042 (1969).
G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, Berlin (1989).
M. B. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York (1972).
P. A. Clarkson, Chaos Solitons Fractals, 5, 2261–2301 (1995).
P. A. Clarkson and E. L. Mansfield, SIAM J. Appl. Math., 54, 1693–1719 (1994); arXiv:solv-int/9409003v1 (1994).
N. Bilă and J. Niesen, J. Symbolic Comput., 38, 1523–1533 (2004).
M. S. Bruzón and M. L. Gandarias, Commun. Nonlinear Sci. Numer. Simul., 13, 517–523 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 168, No. 1, pp. 24–34, July, 2011.
Rights and permissions
About this article
Cite this article
Bruzón, M.S., Gandarias, M.L. Classical and nonclassical symmetries for the Krichever-Novikov equation. Theor Math Phys 168, 875–885 (2011). https://doi.org/10.1007/s11232-011-0071-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-011-0071-5