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Semidirect sums of Lie algebras and discrete integrable couplings. (English) Zbl 1111.37059
Summary: A relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semidirect sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.

MSC:
37K60 Lattice dynamics; integrable lattice equations
17B80 Applications of Lie algebras and superalgebras to integrable systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
81R12 Groups and algebras in quantum theory and relations with integrable systems
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References:
[1] DOI: 10.1016/0960-0779(95)00104-2 · Zbl 1080.37578 · doi:10.1016/0960-0779(95)00104-2
[2] Ma W. X., Methods Appl. Anal. 7 pp 21– (2000) · Zbl 1001.37061 · doi:10.4310/MAA.2000.v7.n1.a2
[3] DOI: 10.1016/0375-9601(96)00112-0 · Zbl 1073.37537 · doi:10.1016/0375-9601(96)00112-0
[4] DOI: 10.1016/S0375-9601(03)01137-X · Zbl 1042.37057 · doi:10.1016/S0375-9601(03)01137-X
[5] DOI: 10.1063/1.1845971 · Zbl 1067.37096 · doi:10.1063/1.1845971
[6] DOI: 10.1063/1.1623000 · Zbl 1063.37068 · doi:10.1063/1.1623000
[7] DOI: 10.1016/j.chaos.2003.10.017 · Zbl 1048.37063 · doi:10.1016/j.chaos.2003.10.017
[8] Frappat L., Dictionary on Lie Algebras and Superalgebras (2000) · Zbl 0965.17001
[9] Yu. Sakovich S., J. Nonlinear Math. Phys. 5 pp 230– (1998) · Zbl 0946.35092 · doi:10.2991/jnmp.1998.5.3.1
[10] Yu. Sakovich S., J. Nonlinear Math. Phys. 6 pp 255– (1999) · Zbl 0947.35142 · doi:10.2991/jnmp.1999.6.3.2
[11] DOI: 10.1016/S0378-4371(00)00592-6 · Zbl 0978.35050 · doi:10.1016/S0378-4371(00)00592-6
[12] Ma W. X., Chaos, Solitons Fractals 13 pp 1451– (2002) · Zbl 1067.37097 · doi:10.1016/S0960-0779(01)00152-7
[13] DOI: 10.1063/1.1432775 · Zbl 1059.37052 · doi:10.1063/1.1432775
[14] Fan G., Chaos, Solitons Fractals 25 pp 425– (2005) · Zbl 1092.37044 · doi:10.1016/j.chaos.2004.08.010
[15] Ma W. X., Phys. Lett. A 351 pp 125– (2006) · Zbl 1234.37049 · doi:10.1016/j.physleta.2005.09.087
[16] DOI: 10.1088/0305-4470/23/17/020 · Zbl 0717.58027 · doi:10.1088/0305-4470/23/17/020
[17] DOI: 10.1063/1.532872 · Zbl 0984.37097 · doi:10.1063/1.532872
[18] DOI: 10.1063/1.522558 · Zbl 0296.34062 · doi:10.1063/1.522558
[19] DOI: 10.1088/0305-4470/37/4/018 · Zbl 1075.37030 · doi:10.1088/0305-4470/37/4/018
[20] Tam H. W., Chaos, Solitons Fractals 23 pp 151– (2005) · Zbl 1075.37027 · doi:10.1016/j.chaos.2004.04.003
[21] Yang H. X., Chaos, Solitons Fractals 26 pp 1091– (2005) · Zbl 1081.37040 · doi:10.1016/j.chaos.2005.02.011
[22] DOI: 10.1063/1.532537 · Zbl 0933.35176 · doi:10.1063/1.532537
[23] DOI: 10.1088/0305-4470/32/11/016 · Zbl 0941.35112 · doi:10.1088/0305-4470/32/11/016
[24] Zhang D. J., Chaos, Solitons Fractals 14 pp 573– (2002) · Zbl 1067.37114 · doi:10.1016/S0960-0779(01)00238-7
[25] Toda M., Springer Series in Solid-State Sciences 20, in: Theory of Nonlinear Lattices, 2. ed. (1989) · doi:10.1007/978-3-642-83219-2
[26] Ma W. X., Int. J. Theor. Phys. 43 pp 219– (2004) · Zbl 1058.37055 · doi:10.1023/B:IJTP.0000028860.27398.a1
[27] Manakov S. V., Sov. Phys. JETP 40 pp 269– (1975)
[28] DOI: 10.1103/PhysRevB.9.1924 · Zbl 0942.37504 · doi:10.1103/PhysRevB.9.1924
[29] DOI: 10.1007/BF01617919 · doi:10.1007/BF01617919
[30] DOI: 10.1016/0370-1573(81)90023-5 · doi:10.1016/0370-1573(81)90023-5
[31] Ma W. X., Chaos, Solitons Fractals 22 pp 395– (2004) · Zbl 1090.37053 · doi:10.1016/j.chaos.2004.02.011
[32] DOI: 10.1016/j.physa.2004.06.072 · doi:10.1016/j.physa.2004.06.072
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