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Discontinuity, Nonlinearity, and Complexity 3(4) (2014) 427--434 | DOI:10.5890/DNC.2014.12.005

Yufeng Zhang$^{1}$, Wenjuan Rui$^{1}$,Hon-Wah 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 6-dimensional 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 Davey-Stewartson (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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