Tri-integrable couplings of the Giachetti-Johnson soliton hierarchy as well as their Hamiltonian structure. (English) Zbl 1474.37087

Summary: Based on zero curvature equations from semidirect sums of Lie algebras, we construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy of soliton equations and establish Hamiltonian structures of the resulting tri-integrable couplings by the variational identity.


37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B80 Applications of Lie algebras and superalgebras to integrable systems
35Q51 Soliton 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] Ma, W. X.; Fuchssteiner, B., Integrable theory of the perturbation equations, Chaos, Solitons & Fractals, 7, 8, 1227-1250 (1996) · Zbl 1080.37578
[2] Zhang, Y. F.; Zhang, H. Q., A direct method for integrable couplings of TD hierarchy, Journal of Mathematical Physics, 43, 1, 466-472 (2002) · Zbl 1052.37055
[3] You, F. C., Nonlinear super integrable couplings of super Dirac hierarchy and its super Hamiltonian structures, Communications in Theoretical Physics, 57, 6, 961-966 (2012) · Zbl 1247.37071
[4] Yu, F. J.; Zhang, H. Q., Hamiltonian structure of the integrable couplings for the multicomponent Dirac hierarchy, Applied Mathematics and Computation, 197, 2, 828-835 (2008) · Zbl 1134.37027
[5] Ma, W. X.; Gao, L., Coupling integrable couplings, Modern Physics Letters B: Condensed Matter Physics, Statistical Physics, Applied Physics, 23, 15, 1847-1860 (2009) · Zbl 1168.37320
[6] Xia, T. C.; Chen, X. H.; Chen, D. Y., A new Lax integrable hierarchy, \(N\) Hamiltonian structure and its integrable couplings system, Chaos, Solitons & Fractals, 23, 2, 451-458 (2005) · Zbl 1071.37047
[7] Xu, X. X., Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures, Journal of Physics A: Mathematical and Theoretical, 42, 39 (2009) · Zbl 1190.37077
[8] Yu, F. J.; Li, L., A new method to construct integrable coupling system for Burgers equation hierarchy by Kronecker product, Communications in Theoretical Physics, 51, 1, 23-26 (2009) · Zbl 1177.35186
[9] You, F. C., Nonlinear super integrable Hamiltonian couplings, Journal of Mathematical Physics, 52, 12 (2011) · Zbl 1273.81150
[10] You, F. C.; Xia, T. C., The integrable couplings of the generalized coupled Burgers hierarchy and its Hamiltonian structures, Chaos, Solitons & Fractals, 36, 4, 953-960 (2008) · Zbl 1142.37047
[11] Ma, W. X.; Fuchssteiner, B., The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Physics Letters A, 213, 1-2, 49-55 (1996) · Zbl 1073.37537
[12] Ma, W., Enlarging spectral problems to construct integrable couplings of soliton equations, Physics Letters A, 316, 1-2, 72-76 (2003) · Zbl 1042.37057
[13] Zhang, Y., A generalized multi-component Glachette-Johnson (GJ) hierarchy and its integrable coupling system, Chaos, Solitons and Fractals, 21, 2, 305-310 (2004) · Zbl 1048.37063
[14] Yu, F.; Li, L., A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product, Physics Letters A, 372, 20, 3548-3554 (2008) · Zbl 1220.35146
[15] Yu, F., A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure, Physics Letters A, 375, 13, 1504-1509 (2011) · Zbl 1242.37051
[16] Ma, W., Loop algebras and bi-integrable couplings, Chinese Annals of Mathematics B, 33, 2, 207-224 (2012)
[17] Meng, J.; Ma, W., Hamiltonian tri-integrable couplings of the AKNS hierarchy, Communications in Theoretical Physics, 59, 4, 385-392 (2013) · Zbl 1264.37026
[18] Ma, W. X.; Meng, J. H.; Zhang, H. Q., Tri-integrable couplings by matrix loop algebras · Zbl 1401.37077
[19] Guo, F.; Zhang, Y., A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling, Journal of Mathematical Physics, 44, 12, 5793-5803 (2003) · Zbl 1063.37068
[20] Fan, E.; Zhang, Y., A simple method for generating integrable hierarchies with multi-potential functions, Chaos, Solitons and Fractals, 25, 2, 425-439 (2005) · Zbl 1092.37044
[21] Xia, T.; Yu, F.; Zhang, Y., The multi-component coupled Burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions, Physica A: Statistical Mechanics and Its Applications, 343, 1-4, 238-246 (2004)
[22] Li, Z.; Dong, H., Two integrable couplings of the Tu hierarchy and their Hamiltonian structures, Computers & Mathematics with Applications, 55, 11, 2643-2652 (2008) · Zbl 1142.37360
[23] Wang, X.; Fang, Y.; Dong, H., Component-trace identity for Hamiltonian structure of the integrable couplings of the Giachetti-Johnson (GJ) hierarchy and coupling integrable couplings, Communications in Nonlinear Science and Numerical Simulation, 16, 7, 2680-2688 (2011) · Zbl 1221.37136
[24] Zhang, Y.; Fan, E., Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti-Pempinelli-Tu hierarchy, Journal of Mathematical Physics, 51, 8 (2010) · Zbl 1312.35156
[25] Ma, W., Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Analysis, 71, 12, e1716-e1726 (2009) · Zbl 1238.37020
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