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The fractional quadratic-form identity and Hamiltonian structure of an integrable coupling of the fractional Ablowitz-Kaup-Newell-segur hierarchy. (English) Zbl 1298.37062

Summary: Starting from a general isospectral problem of fractional order, we propose a fractional quadratic-form identity, from which the Hamiltonian structure of an integrable coupling of the fractional Ablowitz-Kaup-Newell-Segur hierarchy is derived. The method can be generalized to other fractional integrable couplings.{
©2013 American Institute of Physics}

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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[1] Zaslavsky, G. M., Phys. Rep., 371, 461 (2002) · Zbl 0999.82053
[2] Tarasov, V. E.; Zaslavsky, G. M., Physica A, 354, 249 (2005)
[3] Tarasov, V. E., Chaos, 14, 123 (2004)
[4] Tarasov, V. E., Phys. Rev. E, 71, 011102 (2005)
[5] Tarasov, V. E., J. Phys.: Conf. Ser., 7, 17 (2005)
[6] Nigmatullin, R., Phys. Status Solidi B, 133, 425 (1986)
[7] Wu, G. C., Commun. Frac. Calc., 2, 27 (2011)
[8] Zhang, J.; You, F. C., Commun. Frac. Calc., 2, 36 (2011)
[9] Metzler, R.; Klafter, J., J. Phys. A, 37, R161 (2004) · Zbl 1075.82018
[10] Liu, J. C.; Hou, G. L., Chin. Phys. B., 19, 110203 (2010)
[11] Zhou, S. B.; Lin, X. R.; Li, H., Commun. Nonlinear Sci. Numer. Simul., 16, 1533 (2011)
[12] Qi, D. L.; Wang, Q.; Yang, J., Chin. Phys. B., 20, 100505 (2011)
[13] Ge, H. X.; Liu, Y. Q.; Cheng, R. J., Chin. Phys. B., 21, 010206 (2012)
[14] Baleanu, D.; Guvenc, Z. B.; Machado, J. A., Tenreiro (2009)
[15] Podlubny, I., Fractiional Differential Equtions (1999)
[16] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives Theory and Applications (1993) · Zbl 0818.26003
[17] Zaslavsky, G. M., Hamitonian Chaos and Fractional Dynamics (2005)
[18] Riewe, F., Phys. Rev. E, 53, 1890 (1996)
[19] Riewe, F., Phys. Rev. E, 55, 3581 (1997)
[20] Frederico, G. S. F.; Torres, D. F. M., Nonlinear Dyn., 53, 215 (2008) · Zbl 1170.49017
[21] Wu, G. C., Commun. Frac. Calc., 1, 27 (2010)
[22] Baleanu, D.; Muslih, S. I., Phys. Scr., 72, 119 (2005) · Zbl 1122.70360
[23] Baleanu, D.; Agrawal, O. P., Czech J. Phys., 56, 1087 (2006) · Zbl 1111.37304
[24] El-Nabulsi, R. A., Fizika, 14, 289 (2005)
[25] El-Nabulsi, R. A., Int. J. Appl. Math. Comput., 17, 299 (2005)
[26] El-Nabulsi, R. A.; Torres, D. F. M., J. Math. Phys., 49, 053521 (2008) · Zbl 1152.81422
[27] Agrawal, O. P., J. Phys. A: Math. Theor., 39, 10375 (2006) · Zbl 1097.49021
[28] Agrawal, O. P., J. Phys. A: Math. Theor., 40, 6287 (2007) · Zbl 1125.26007
[29] Tarasov, V. E., J. Phys. A: Math. Gen., 39, 8409 (2006) · Zbl 1122.70013
[30] Jumarie, G., Chaos Solitons Fractals, 32, 969 (2007) · Zbl 1154.70011
[31] Almeida, R.; Malinowska, A. B.; Torres, D. F. M., J. Math. Phys., 51, 033503 (2010) · Zbl 1309.49003
[32] Malinowska, A. B.; Ammi, M. R. S.; Torres, D. F. M., Commun. Frac. Calc., 1, 32 (2010)
[33] Tu, G. Z., J. Math. Phys., 30, 330 (1989) · Zbl 0678.70015
[34] Ma, W. X., Chin. Ann. Math., Ser. A, 18, 115 (1992)
[35] Zhang, Y. F.; Zhang, H. Q., J. Math. Phys., 43, 466 (2002) · Zbl 1052.37055
[36] Xia, T. C.; You, F. C,; Chen, D. Y., Chaos, Solitons Fractals, 23, 1911 (2005)
[37] Ma, W. X.; Chen, M. J., Phys. A: Math. Gen., 39, 10787 (2006) · Zbl 1104.70011
[38] Ma, W. X.; Xu, X. X.; Zhang, Y. F. J., Math. Phys., 47, 053501 (2006) · Zbl 1111.37059
[39] Wu, G. C.; Zhang, S., Phys. Lett. A, 375, 3659 (2011) · Zbl 1252.37063
[40] Kolwankar, K. M.; Gangal, A. D., Chaos, 6, 505 (1996) · Zbl 1055.26504
[41] Kolwankar, K. M.; Gangal, A. D., Pramana, J. Phys., 48, 49 (1997)
[42] Kolwankar, K. M.; Gangal, A. D., Phys. Rev. Lett., 80, 214 (1998) · Zbl 0945.82005
[43] Chen, W., Chaos Solitons Fractals, 28, 923 (2006) · Zbl 1098.60078
[44] Chen, W.; Sun, H. G., Mod. Phys. Lett. B, 23, 449 (2009) · Zbl 1386.76089
[45] Cresson, J., J. Math. Phys., 44, 4907 (2003) · Zbl 1062.39022
[46] Cresson, J., J. Math. Anal. Appl., 307, 48 (2005) · Zbl 1077.49033
[47] Parvate, A.; Gangal, A. D., Fractals, 17, 53 (2009) · Zbl 1173.28005
[48] Jumarie, G., Comput. Math. Appl., 51, 1367 (2006) · Zbl 1137.65001
[49] Li, X., Davison, M., and Essex, C., “On the concept of local fractional differentiation,” preprint, see .
[50] Chen, Y.; Yan, Y.; Zhang, K. W., J. Math. Anal. Appl., 362, 17 (2010) · Zbl 1196.26011
[51] Yang, X. J., Research on fractal mathematics and some applications in mechanics (2009)
[52] Adda, F. B., J. Fractional Calculus, 11, 21 (1997)
[53] Adda, F. B.; Cressonc, J., J. Math. Anal. Appl., 263, 721 (2001) · Zbl 0995.26006
[54] Cottrill-Shepherd, K.; Naber, M., J. Math. Phys., 42, 2203 (2001) · Zbl 1011.58001
[55] Kazbekov, K. K., Vladikavkaz Mat. Zh., 7, 2, 41 (2005)
[56] Li, C. P.; Deng, W. H., Appl Math. Comput., 187, 777 (2007) · Zbl 1125.26009
[57] Guo, F. K.; Zhang, Y. F., J. Phys. A, 38, 8537 (2005) · Zbl 1077.37045
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