Abstract
An algorithmic method to construct a kind of auto Bäcklund transformations (BTs) is proposed. A Maple package AutoBT, which can entirely automatically generate auto BT is presented. AutoBT has been effectively applied to many nonlinear evolution equations with physical significance. Not only are previously known BT recovered but also in some cases new and more general form of BT are obtained.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Steeb, W.H., Grauel, A., Kloke, M., Spieker, B.M.: Nonlinear diffusion equations, integrability and the Painleve property. Phys. Scripta. 31, 5 (1985)
Xu, G.Q., Li, Z.B.: A maple package for the Painleve test of nonlinear partial differential equations. Chin. Phys. Lett. 20, 975 (2003)
Lou, S.Y.: Painleve test for the integrable dispersive long waves. Phys. Lett. A 176, 96 (1993)
Wang, M.L., Zhou, Y.B., Li, Z.B.: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A 216, 67 (1996)
Fan, E.G.: Two new applications of the homogeneous balance method. Phys. Lett. A 265, 353 (2000)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
Ma, W.X., Geng, X.G.: Bäcklund transformations of soliton systems from symmetry constraints. In: Bäcklund and Darboux Transformations. The Geometry of Solitons. CRM Proceedings & Lecture Notes, vol. 29. American Mathematics Society (2001)
Lin, J., Lou, S.Y.: Multisoliton solutions of the (3+1)-dimensional Nizhnik-Novikov-Veselov equation. Comm. Theor. Phys. 37, 265–268 (2002)
Van de Leur, J.: Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra. J. Geo. Phys. 57, 435–447 (2007)
Sokalski, K., Wietecha, T., Lisowski, Z.: Variational approach to the Bäcklund transformations. Acta Physica Polonica B 32, 17–28 (2001)
Sokalski, K., Wietecha, T., Sokalska, D.: Existence of dual equations by means of strong necessary conditions - Analysis of integrability of partial differential nonlinear equations. J. Non. Math. Phys. 12, 31–52 (2005)
Fu, Z.T., Liu, S.K., Liu, S.D.: New kinds of solutions to Gardner equation. Chaos, Solitons and Fractals 20, 301–309 (2004)
Kichenassamy, S., Oliver, P.J.: Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23, 1141–1166 (1992)
Liu, S.K., Liu, S.D.: Nonlinear Equations in Physics. Peking University Press, Beijing (2000)
Morrison, A.J., Parks, E.J.: The N-soliton solution of the modified generalised Vakhnenko equation. Chaos, Solitons and Fractals 16, 13–26 (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Li, Z., Liu, Y., Qian, H. (2008). A Method and Its Implementation for Constructing Bäcklund Transformations to Nonlinear Evolution Equations. In: Kapur, D. (eds) Computer Mathematics. ASCM 2007. Lecture Notes in Computer Science(), vol 5081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87827-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-87827-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87826-1
Online ISBN: 978-3-540-87827-8
eBook Packages: Computer ScienceComputer Science (R0)