Discontinuity, Nonlinearity, and Complexity
An (2+1)dimensional Expanding Model of the DaveyStewartson Hierarchy As Well As Its Hamiltonian Structure
Discontinuity, Nonlinearity, and Complexity 3(4) (2014) 427434  DOI:10.5890/DNC.2014.12.005
Yufeng Zhang$^{1}$, Wenjuan Rui$^{1}$,HonWah Tam$^{3}$
$^{1}$ College of Sciences, China University of Mining and Technology, Xuzhou 221116, P.R. China
$^{2}$ Department of Computer Science, Hong Kong Baptist University, Hong Kong, P.R. China
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Abstract
Introducing a new 6dimensional Lie algebra aims at generating a Lax pair whose compatibility condition gives rise to (1+1)dimensional integrable hierarchy of equations which can reduce to the nonlinear Schr¨odinger equation and two sets of nonlinear integrable equations by taking various parameters. The Hamiltonian structure of the (1+1)dimensional hierarchy is also obtained by using the trace identity. The reason for generating the above (1+1)dimensional integrable hierarchy lies in obtaining (2+1)dimensional equation hierarchy. That is to say, with the hep of the higher dimensional Lie algebra, we introduce two 4 × 4 matrix operators in an associative algebra A [ ξ ] for which a new (2+1)dimensional hierarchy of equations is derived by using the TAH scheme and the Hamiltonian operator in the case of 1+1 dimensions , which generalizes the results presented by Tu, that is, the reduced case of the hierarchy obtained by us can be reduced to the DaveyStewartson (DS) hierarchy. Finally, the Hamiltonian structure of the (2+1)dimensional hierarchy is produced by the trace identity used for 2+1 dimensions, which was proposed by Tu. As we have known that there is no paper involving such the problem on generating expanding models of (2+1)dimensional integrable hierarchy.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities(2013XK03) and the National Natural Science Foundation of China (grant No. 11371361).
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