A few new higher-dimensional Lie algebras and two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy. (English) Zbl 1221.37143

Zhang, Yu-Feng; Tam, Honwah

Commun. Nonlinear Sci. Numer. Simul. 16, No. 1, 76-85 (2011).

Summary: Four higher-dimensional Lie algebras are introduced. With the help of their different loop algebras and the block matrices of Lax pairs for the zero curvature representations of two given integrable couplings, the two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively, which fill up the consequences obtained by W. Ma and L. Gao [Mod. Phys. Lett. B 23, No. 15, 1847–1860 (2009; Zbl 1168.37320)]. The coupling integrable couplings of the AKNS hierarchy obtained in the paper again reduce to the coupling integrable couplings of the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, which are different from the resulting results given in [loc. cit.].

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures

Keywords:

Lie algebra; integrable couplings; zero curvature equation

Citations:

Zbl 1168.37320

References:

[1]	Ma, W. X.; Fuchssteiner, B., Chaos Soliton Fract, 7, 1227 (1996)
[2]	Ma, W. X., Meth Appl Anal, 7, 21 (2000)
[3]	Ma, W. X.; Fuchssteiner, B., Phys Lett A, 213, 49 (1996)
[4]	Guo, F. K.; Zhang, Y. F., J Math Phys, 44, 5793 (2003)
[5]	Zhang, Y. F., Chaos Soliton Fract, 21, 305 (2004)
[6]	Xia, T. C.; Chen, X. H.; Chen, D. Y., Chaos Soliton Fract, 23, 451 (2005)
[7]	Li, Z.; Zhang, Y. J.; Dong, H. H., Mod Phys Lett B, 21, 595 (2007)
[8]	Yu, F. J.; Zhang, H. Q., Appl Math Comput, 197, 828 (2008)
[9]	Ma, W. X.; Gao, L., Mod Phys Lett B, 23, 15, 1847 (2009)
[10]	Ma, W. X., Phys Lett A, 35, 125 (2006)
[11]	Das, A., Integrable models (1989), World Scientific Publishing: World Scientific Publishing Teaneck (NJ) · Zbl 0719.70011
[12]	Tu, G. Z., J Math Phys, 30, 2, 330 (1989)
[13]	Ma, W. X., Chin J Contemp Math, 13, 79 (1992)
[14]	Guo, F. K.; Zhang, Y. F., J Phys A, 38, 8537 (2005)
[15]	Ma, W. X.; Chen, M., J Phys A, 39, 10787 (2006)
[16]	Zhang, Y. F.; Dong, H. H.; Tam, H., Commun Theor Phys (Beijing, China), 48, 215 (2007)
[17]	Cao, C. W.; Wu, Y. T.; Geng, X. G., J Math Phys, 8, 3948 (1999)
[18]	Geng, X. G.; Dai, H. H., J Math Phys, 41, 337 (2000)
[19]	Fan, E. G., J Phys A, 42, 9, 95206 (2009)
[20]	Zeng, Y. B., Chin Sci Bull, 37, 1937 (1992)
[21]	Zeng, Y. B., J Math Phys, 38, 321 (1997)