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IMA Journal of Applied Mathematics Advance Access originally published online on October 23, 2008
IMA Journal of Applied Mathematics 2009 74(1):46-61; doi:10.1093/imamat/hxn024
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Symbolic-computation study of integrable properties for the (2 + 1)-dimensional Gardner equation with the two-singular manifold method

Hai-Qiang Zhang

School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China

Bo Tian{dagger}

School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China, State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100083, China and Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China

Juan Li, Tao Xu and Ya-Xing Zhang

School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China

{dagger} Email: gaoyt{at}public.bta.net.cn

Received on September 29, 2007; Revision received September 29, 2007. Accepted on July 23, 2008

The singular manifold method from the Painlevé analysis can be used to investigate many important integrable properties for the non-linear partial differential equations. In this paper, the two-singular manifold method is applied to the (2 + 1)-dimensional Gardner equation with two Painlevé expansion branches to determine the Hirota bilinear form, Bäcklund transformation, Lax pairs and Darboux transformation. Based on the obtained Lax pairs, the binary Darboux transformation is constructed and the N x N Grammian solution is also derived by performing the iterative algorithm N times with symbolic computation.

Keywords: (2 + 1)-dimensional Gardner equation; Bäcklund transformation; Darboux transformation; soliton; Painevé analysis.


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