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A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling. (English) Zbl 1063.37068
Summary: A type of new interesting loop algebra \(\widetilde G_M\), \(M = 1,2,\dots\), with a simple commutation operation just like that in the loop algebra \(\widetilde A_1\) is constructed. With the help of the loop algebra \(\widetilde G_M\), a new multicomponent integrable system, M-AKNS-KN hierarchy, is worked out. As reduction cases, the M-AKNS hierarchy and M-KN hierarchy are engendered, respectively. In addition, the system 1-AKNS-KN, which is a reduced case of the M-AKNS-KN hierarchy above, is a unified expressing integrable model of the AKNS hierarchy and the KN hierarchy. Obviously, the M-AKNS-KN hierarchy is again a united expressing integrable model of the multicomponent AKNS hierarchy (M-AKNS) and the multicomponent KN hierarchy(M-KN). This article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations. Finally, we work out an integrable coupling of the M-AKNS-KN hierarchy.

37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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[1] M. J. Ablowitz and P. A. Clarkson,Solitons, Nonlinear Evolution Equations and Inverse Scattering(Cambridge University Press, Cambridge, 1991). · Zbl 0762.35001
[2] A. C. Newell,Soliton in Mathematics and Physics(SIAM, Philadelphia, 1985).
[3] Wadati, J. Phys. Soc. Jpn. 32 pp 1447– (1972)
[4] Wadati, J. Phys. Soc. Jpn. 34 pp 1289– (1973)
[5] Wadati, Prog. Theor. Phys. 53 pp 417– (1975)
[6] Wadati, J. Phys. Soc. Jpn. 47 pp 1698– (1979)
[7] Shimizu, Prog. Theor. Phys. 63 pp 808– (1980)
[8] Tu, J. Math. Phys. 30 pp 330– (1989)
[9] Guo, Acta Math. Phys. Sin. 19 pp 507– (1999)
[10] Tsuchida, J. Phys. Soc. Jpn. 69 pp 2241– (1999)
[11] Tsuchida, Phys. Lett. A 257 pp 53– (1999)
[12] Guo, Acta Math. Appl. Sin. 23 pp 181– (2000)
[13] Guo, J. Syst. Sci. Math. Sci. 22 pp 36– (2003)
[14] Ma, Chin. Ann. Math., Ser. A 13 pp 115– (1992)
[15] Ma, Chin. Ann. Math., Ser. B 23 pp 373– (2002)
[16] F. Guo and Y. Zhang, Chaos, Solitons Fractals, 2003 (accepted for publication).
[17] Guo, Acta Phys. Sin. 51 pp 951– (2002)
[18] Ma, Chaos, Solitons Fractals 7 pp 1227– (1996)
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