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A generalized AKNS hierarchy and its bi-Hamiltonian structures. (English) Zbl 1071.37048

Summary: First, we construct a new isospectral problem with 8 potentials. And then, a new Lax pair is presented. By making use of Tu’s scheme, a class of new soliton hierarchies of equations is derived, which is integrable in the sense of Liouville and possesses bi-Hamiltonian structures. After making some reductions, the well-known AKNS hierarchy and other hierarchies of evolution equations are obtained. Finally, in order to illustrate that the soliton hierarchy obtained in the paper possesses bi-Hamiltonian structures exactly, we prove that the linear combination of two Hamiltonian operators admitted is also a Hamiltonian operator constantly. We point out that two Hamiltonian operators obtained of the system are directly derived from a recurrence relation, not from a recurrence operator.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, nonlinear evolution equations and inverse scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[2] Newell, A. C., Soliton in mathematics and physics (1985), SIAM: SIAM Philadelphia
[3] Wadati, M., J. Phys. Soc. Jan., 32, 1681 (1972)
[4] Wadati, M., J. Phys. Soc. Jan., 34, 1289-1296 (1973)
[5] Wadati, M.; Sanuki, H.; Konno, K., Prog. Theor. Phys., 53, 419-436 (1975)
[6] Tu, G. Z., J. Math. Phys., 30, 330-337 (1989)
[7] Fan, E. G., J. Phys. A: Math. Gen., 34, 513-519 (2001)
[8] Yan, Z. Y.; Zhang, H. Q., Chaos, Solitons & Fractals, 13, 1439-1450 (2002), 2002;14:1445-56; 2002;14:441-6
[9] Yan, Z. Y., Chaos, Solitons & Fractals, 15, 639-645 (2003)
[10] Fan, E. G., Physica A, 301, 105-110 (2001)
[11] Fan, E. G., J. Math. Phys., 41, 7769-7778 (2000)
[12] Tsuchida, T.; Wadati, M., Chaos, Solitons & Fractals, 9, 869-873 (1998)
[13] Ma, W. X., Chaos, Solitons & Fractals, 7, 1227-1250 (1996)
[14] Ma, W. X., Chin. Math. Ann., 13, 115-123 (1992), [in Chinese]
[15] Ma, W. X.; Zhou, R. G., A coupled AKNS-Kaup-Newell soliton hierarchy, J. Math. Phys., 40, 4419-4428 (1999) · Zbl 0947.35118
[16] Zhang, Y. F., Chaos, Solitons & Fractals, 18, 855-862 (2003), 2004;21:305-10
[17] Zhang, Y. F.; Yan, Q. Y., Chaos, Solitons & Fractals, 21, 413-423 (2004)
[18] Xu, X. X., Chaos, Solitons & Fractals, 15, 475-486 (2003)
[19] Cao, C. W., Sci. China. Ser. A, 33, 528-536 (1990)
[20] Geng, X. G., J. Math. Phys., 34, 805-817 (1993)
[21] Zeng, Y. B., Phys. Lett. A, 160, 541-547 (1991)
[22] Zeng, Y. B.; Hietarinta, J., J. Phys. A: Math. Gen., 29, 5241-5251 (1996)
[23] Kundu, A., Physica D, 25, 339 (1987)
[24] Qiao, Z., J. Phys. A: Math. Gen., 26, 4407-4417 (1993) · Zbl 0802.35135
[25] Xia, T.; Chen, X.; Chen, D., Chaos, Solitons & Fractals, 22, 939-945 (2004)
[26] Xia, T.; Chen, X.; Chen, D., Chaos, Solitons & Fractals, 23, 451-458 (2005)
[27] Guo, F., Acta Math. Appl. Sinaca, 23, 2, 181 (2000)
[28] Guo, F., J. Sys. Sci. Math. Sci., 22, 1, 36 (2002)
[29] Xia, T.; Chen, X.; Chen, D.; Zhang, Y. F., Commun. Theor. Phys. (Beijing, China), 42, 180-182 (2004)
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