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Nonlinear continuous integrable Hamiltonian couplings. (English) Zbl 1234.37047
In this paper, a scheme is established to construct nonlinear continuous integrable couplings by means of a kind of special non-semisimple matrix Lie algebras. Variational identities over the associated loop algebras are used to build Hamiltonian structures for the resulting couplings. The AKNS hierarchy of soliton equations is used to give an example of application of the scheme.

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35A30 Geometric theory, characteristics, transformations in context of PDEs
35F50 Systems of nonlinear first-order PDEs
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
Full Text: DOI
[1] Ma, W.X.; Fuchssteiner, B., Integrable theory of the perturbation equations, Chaos solitons fract., 7, 1227-1250, (1996) · Zbl 1080.37578
[2] Ma, W.X., Integrable couplings of soliton equations by perturbations. I - A general theory and application to the KdV hierarchy, Methods appl. anal., 7, 21-55, (2000) · Zbl 1001.37061
[3] Ma, W.X.; Xu, X.X.; Zhang, Y.F., Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. lett. A, 351, 125-130, (2006) · Zbl 1234.37049
[4] Ma, W.X.; Xu, X.X.; Zhang, Y.F., Semidirect sums of Lie algebras and discrete integrable couplings, J. math. phys., 47, 053501, (2006), 16 pp · Zbl 1111.37059
[5] Ma, W.X.; Chen, M., Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. phys. A: math. gen., 39, 10787-10801, (2006) · Zbl 1104.70011
[6] Ma, W.X., A discrete variational identity on semi-direct sums of Lie algebras, J. phys. A: math. theoret., 40, 15055-15069, (2007) · Zbl 1128.22014
[7] Ma, W.X., A bi-Hamiltonian formulation for triangular systems by perturbations, J. math. phys., 43, 1408-1421, (2002) · Zbl 1059.37052
[8] Ma, W.X., Enlarging spectral problems to construct integrable couplings of soliton equations, Phys. lett. A, 316, 72-76, (2003) · Zbl 1042.37057
[9] Xia, T.C.; Yu, F.J.; Zhang, Y., The multi-component coupled Burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions, Physica A, 343, 238-246, (2004)
[10] Ma, W.X., Integrable couplings of vector AKNS soliton equations, J. math. phys., 46, 033507, (2005), 19 pp. · Zbl 1067.37096
[11] Li, Z.; Dong, H.H., Two integrable couplings of the Tu hierarchy and their Hamiltonian structures, Comput. math. appl., 55, 2643-2652, (2008) · Zbl 1142.37360
[12] Zhang, Y.F.; Tam, H.W., Coupling commutator pairs and integrable systems, Chaos solitons fract., 39, 1109-1120, (2009) · Zbl 1197.37086
[13] Xu, X.X., Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures, J. phys. A: math. theoret., 42, 395201, (2009), 21 pp · Zbl 1190.37077
[14] Ma, W.X., Variational identities and Hamiltonian structures, (), 1-27 · Zbl 1230.37086
[15] Ma, W.X.; Zhang, Y., Component-trace identities for Hamiltonian structures, Appl. anal., 89, 457-472, (2010) · Zbl 1192.37091
[16] Drinfel’d, V.G.; Sokolov, V.V., Equations of Korteweg-de Vries type and simple Lie algebras, Soviet math. dokl., 23, 457-462, (1981) · Zbl 0513.35073
[17] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. appl. math., 53, 249-315, (1974) · Zbl 0408.35068
[18] Zakharov, V.E.; Shabat, A.B., A scheme for generating the nonlinear equations of mathematical physics by the method of the inverse scattering problem I, Funct. anal. appl., 8, 1974, 226-235, (1975) · Zbl 0303.35024
[19] Zhang, Y.F.; Tam, H.W., Applications of the Lie algebra gl(2), Modern phys. lett. B, 23, 1763-1770, (2009) · Zbl 1167.37356
[20] Tu, G.Z., On Liouville integrability of zero-curvature equations and the Yang hierarchy, J. phys. A: math. gen., 22, (1989)
[21] Magri, F., A simple model of the integrable Hamiltonian equation, J. math. phys., 19, 1156-1162, (1978) · Zbl 0383.35065
[22] Fuchssteiner, B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear anal., 3, 849-862, (1979) · Zbl 0419.35049
[23] Ma, W.X.; Gao, L., Coupling integrable couplings, Modern phys. lett. B, 23, 1847-1860, (2009) · Zbl 1168.37320
[24] Ma, W.X.; Strampp, W., Bilinear forms and Bäcklund transformations of the perturbation systems, Phys. lett. A, 341, 441-449, (2005) · Zbl 1171.37332
[25] Sun, Y.P.; Zhao, H.Q., New non-isospectral integrable couplings of the AKNS system, Appl. math. comput., 203, 163-170, (2008) · Zbl 1154.37366
[26] Ma, W.X., A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order, Phys. lett. A, 367, 473-477, (2007) · Zbl 1209.37070
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