Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-23T08:12:35.415Z Has data issue: false hasContentIssue false

Generalised (2+1)-dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory

Published online by Cambridge University Press:  07 September 2015

Huanhe Dong
Affiliation:
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
Kun Zhao*
Affiliation:
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
Hongwei Yang
Affiliation:
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
Yuqing Li
Affiliation:
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
*
*Corresponding author. Email address: zhao1990kun@163.com (K. Zhao)
Get access

Abstract

Much attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

Type
Research Article
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Magri, F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19, 11561162 (1978).Google Scholar
[2]Ablowitz, M.J., Chakravarty, S. and Halburd, R.G., Integrable systems and reductions of the self-dual Yang-Mills equations, J. Math. Phys. 44, 31473173 (2003).Google Scholar
[3]Hirota, R., Direct Methods in Soliton Theory, see pp. 157176, Springer (1980)Google Scholar
[4]Tu, G.Z., A new hierarchy of integrable systems and its Hamiltonian structure, Sci. Sin. A 32, 142153 (1989).Google Scholar
[5]Tu, G.Z., The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys. 30, 330338 (1989).Google Scholar
[6]Ma, W.X., A new hierarchy of Liouville integrable generalised Hamiltonian equations and its reduction, Chin. J. Cont. Math. 13, 7989 (1992).Google Scholar
[7]Tu, G.Z. and Ma, W.X., An algebraic approach for extending Hamiltonian operators, J. Part. Diff. Eq. 5, 4356 (1992).Google Scholar
[8]Hu, X.B., A powerful approach to generate new integrable systems, J. Phys. A: Math. Gen. 27, 24972514 (1994).Google Scholar
[9]Guo, F.K. and Zhang, Y.F., The quadratic-form identity for constructing the Hamiltonian structure of integrable systems, J. Phys. A: Math. Gen. 38, 85378548(2005).Google Scholar
[10]Wang, X.R., Fang, Y. and Dong, H.H., Component-trace identity for Hamiltonian structure of the integrable couplings of the Giachetti-Johnson (GJ) hierarchy and coupling integrable couplings, Comm. Nonlinear Sci. Num. Sim. 16, 26802688 (2011).Google Scholar
[11]Yang, H.W., Yin, B.S. and Fang, Y., A class of new Lie lgebra, the corresponding g-mKdV hierarchy and its Hamiltonian structure, Int. J. Theor. Phys. 50, 671681 (2011).Google Scholar
[12]Ma, W.X., He, J.S. and Qin, Z.Y., A supertrace identity and its application to superintegrable systems, J. Math. Phys. 49, 033511 (2008).Google Scholar
[13]Tu, G.Z., A trace identity and its application to integrable systems of 1+2 dimensions, J. Phys. A: Math. Gen. 32, 19001907 (1991).Google Scholar
[14]Zhang, Y.F. and Rui, W.J., On generating (2+1)-dimensional hierarchies of evolution equations, Comm. Nonlinear Sci. Num. Sim. 19, 34543461 (2014).Google Scholar
[15]Zhang, Y.F., Gao, J. and Wang, G., Two (2+1)-dimensioal hierarchies of evolution equations and their Hamiltonian structures, App. Math. Comp. 243, 601606 (2014).Google Scholar
[16]Wadati, M., Sanuki, H. and Konno, K., Relationships among inverse method, Backlund transformation and an infinite number of conservation laws, Prog. Theor. Phys. 53, 419436 (1975).Google Scholar
[17]Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19, 10951097 (1967).CrossRefGoogle Scholar
[18]Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27, 11921194 (1971).Google Scholar
[19]Malfiet, W., Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60, 650654 (1992).Google Scholar
[20]Yan, C.T., A simple transformation for nonlinear waves, Phys. Lett. A 224, 7784 (1996).CrossRefGoogle Scholar
[21]Gepreel, K.A., A generalised (G’/G)-expansion method to find the traveling wave solutions of nonlinear evolution equations, J. Part. Diff. Eq. 24, 5569 (2011).Google Scholar
[22]Abdeljabbar, A., Ma, W.X. and Yildirim, A., Determinant solutions to a (3+1)-dimensional generalised KP equation with variable coefficients, Chin. Ann. Math. Ser. B 33, 641650 (2012).Google Scholar
[23]Ma, W.X. and Zhu, Z.N., Solving the (3+1)-dimensional generalised KP and BKP equations by the multiple exp-function algorithm, App. Math. Comp. 218, 1187111879 (2012).CrossRefGoogle Scholar
[24]Chen, J.P. and Chen, H., The (g′/g 2)-expansion method and its application to coupled nonlinear Klein-Gordon Equation, J. South China Normal Univ. (Natural Science Edition) 44, 6366 (2012).Google Scholar
[25]Xu, Z.H., Yin, B.S., Hou, Y.J. and Xu, Y.S., Variability of internal tides and near-inertial waves on the continental slope of the northwestern South China Sea, J. Geophys. Res.: Oceans 118, 197211 (2013).Google Scholar
[26]Yang, H.W., Yin, B.S., Dong, H.H. and Ma, Z.D., Generation of solitary Rossby waves by unstable topography, Comm. Theor. Phys. 57, 473476 (2012).Google Scholar
[27]Ono, H., Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39, 10821091 (1975).Google Scholar
[28]Yang, H.W., Wang, X.R. and Yin, B.S., A kind of new algebraic Rossby solitary waves generated by periodic external source, Nonlinear Dyn. 76, 17251735 (2014).Google Scholar