Semi-direct sums of Lie algebras and continuous integrable couplings. (English) Zbl 1234.37049

Ma, Wen-Xiu; Xu, Xixiang; Zhang, Yufeng

Phys. Lett., A 351, No. 3, 125-130 (2006).

Summary: A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems.

MSC:

37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
17B80	Applications of Lie algebras and superalgebras to integrable systems

Keywords:

semi-direct sums of Lie algebras; zero curvature equations; integrable couplings; symmetries

References:

[1]	Ma, W. X.; Fuchssteiner, B., Chaos Solitons Fractals, 7, 1227 (1996)
[2]	Ma, W. X., Methods Appl. Anal., 7, 21 (2000)
[3]	Ma, W. X.; Fuchssteiner, B., Phys. Lett. A, 213, 49 (1996) · Zbl 0863.35106
[4]	Ma, W. X., Phys. Lett. A, 316, 72 (2003)
[5]	Ma, W. X., J. Math. Phys., 46, 033507 (2005)
[6]	Guo, F. G.; Zhang, Y., J. Math. Phys., 44, 5793 (2003)
[7]	Zhang, Y., Chaos Solitons Fractals, 21, 305 (2004)
[8]	Ma, W. X., J. Math. Phys., 43, 1408 (2002)
[9]	Frappat, L.; Sciarrino, A.; Sorba, P., Dictionary on Lie Algebras and Superalgebras (2000), Academic Press: Academic Press San Diego, CA · Zbl 0965.17001
[10]	Sakovich, S. Yu., J. Nonlin. Math. Phys., 5, 230 (1998)
[11]	Sakovich, S. Yu., J. Nonlin. Math. Phys., 6, 255 (1999)
[12]	Ma, W. X.; Zhou, R. G., Physica A, 296, 60 (2001)
[13]	Fan, E. G.; Zhang, Y., Chaos Solitons Fractals, 25, 425 (2005)
[14]	Tu, G. Z., J. Phys. A: Math. Gen., 22, 2375 (1989)
[15]	Ma, W. X., Chinese Ann. Math. Ser. A, 13, 115 (1992)
[16]	Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Studies Appl. Math., 53, 249 (1974)
[17]	Xu, X. X., Chaos Solitons Fractals, 15, 475 (2003)
[18]	Olver, P. J., Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0656.58039
[19]	Ma, W. X.; Strampp, W., Phys. Lett. A, 185, 277 (1994)
[20]	Ma, W. X., J. Phys. A: Math. Gen., 26, 2573 (1993)
[21]	Ma, W. X., J. Math. Phys., 33, 2464 (1992)