ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

An (2+1)-dimensional Expanding Model of the Davey-Stewartson Hierarchy As Well As Its Hamiltonian Structure

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

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).

References

1.  [1] Ablowitz, M.J., Chakravarty, S. and Halburd, R.G. (2003), Integrable systems and reductions of the self-dual Yang- Mills equations, Journal of Mathematical Physics, 44(8), 3147.
2.  [2] Zhang, Y.F., and Hon, Y.C. (2011), Some evolution hierarchies derived from self-dual Yang-Mills equations, Communications in Theoretical Physic, 56, 856.
3.  [3] Chakravarty, S., Kent, S.L., and Newman, E.T. (1996), Some reductions of the self-dual Yang-Mills equations to integrable systems in 2+1 dimensions, Journal of Mathematical Physics, 36, 763.
4.  [4] Tu, G.Z., Andruskiw, R.I., and Huang, X.C. (1991), A trace identity and its application to integrable systems of 1+2 dimensions, Journal of Mathematical Physics, 32, 1990.
5.  [5] Fokas, A.S. and Tu, G.Z. (1990), An algebraic recursion scheme of KP and DS hierarchy, Preprint, Clarkson University.
6.  [6] Tu, G.Z. (1989), The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, Journal of Mathematical Physics, 30, 330.
7.  [7] Fuchssteiner, B. (1993), Coupling of completely integrable system: the perturbation bundle, In: Clarkson PA, ed., Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Kluwer, Dordrecht, 125.
8.  [8] Ma, W.X. (2000), Integrable couplings of soliton equations by perturbation I, A general theory and application to the KdV equation, Mehtods and Applications of Analysis, 7, 21.
9.  [9] Fan, E.G. and Chow, K.W. (2011),Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation, Journal of Mathematical Physics, 52, 023504.