Abstract
In this paper we show that the Darboux transformation for a large class of nonlinear evolution equations arises due to factorization and commutation. The factorization and commutation has been pointed out earlier for Schrödinger operator. We show that it extends to a large class of nonlinear differential equations which admit Lax pairs including Boussinesq, Davey–Stewartson, Bogoyavlensky–Schiff and n-wave interaction equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G Darboux, Compt. Rend. 94, 1456 (1882)
J Poschel and E Trubowitz, Inverse spectral theory (Academic Press, 1987)
V B Matveev and M A Salle, Darboux transformations and solitons (Springer, New York, 1991)
V B Matveev, Lett. Math. Phys. 3(3), 213 (1979)
V B Matveev, Lett. Math. Phys. 3(3), 217 (1979)
V B Matveev, L.D. Faddeev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, pp. 179–209
M Senthilvelan and M Lakshmanan, Nonlinear Math. Phys. 5, 190 (1998)
M S Bruzón et al, Theor. Math. Phys. 137(1), 1367 (2003)
Biao Li and Yong Chen, Czech. J. Phys. 54(5), 517 (2004)
M V Prabhakar and H Bhate, Lett. Math. Phys. 64, 1 (2003)
O I Bogoyavlenskii, Math. USSR Izv. 34, 245 (1990)
J Schiff, Painleve transcendents: Their asymptotics and physical applications (Plenum, New York, 1992)
H C Hu, Phys. Lett. A 373, 1750 (2009)
R Coifman and A S Fokas, Important developments in soliton theory edited by A S Fokas and V E Zakharov (Springer Verlag, 1993) p 58
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
PRABHAKAR, M.V., BHATE, H. Commutation and Darboux transformation. Pramana - J Phys 85, 869–880 (2015). https://doi.org/10.1007/s12043-015-1092-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-015-1092-7