×

Multi-component Volterra and Toda type integrable equations. (English) Zbl 0983.37082

Summary: Multi-component integrable analogs related to the Jordan triple systems (JTS) are constructed for the Volterra equation. Differential-difference substitutions lead to multi-component Toda type lattices. Associated equations generalize the derivative nonlinear Schrödinger equation. Multi-component master symmetries (both partial differential and differential difference ones) and zero curvature representations for lattice equations written in terms of the superstructure Lie algebra of the JTS arise for the first time.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K60 Lattice dynamics; integrable lattice equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Svinolupov, S. I., Phys. Lett. A, 135, 32 (1989)
[2] Svinolupov, S. I., Commun. Math. Phys., 143, 559 (1992)
[3] Habibullin, I. T.; Sokolov, V. V.; Yamilov, R. I., Multi-component integrable systems and nonassociative structures, (Alfinito, E.; Boiti, M.; Martina, L.; Pempinelli, F., Proc. Workshop on Nonlinear Physics: Theory and Experiment (1996), World Scientific: World Scientific Singapore), 139 · Zbl 0941.37523
[4] Shabat, A. B.; Yamilov, R. I., Phys. Lett. A, 130, 271 (1988)
[5] Shabat, A. B.; Yamilov, R. I., Algebra i Analiz. Algebra i Analiz, Leningrad Math. J, 2, 2, 377 (1991), English Transl. in:
[6] Kaup, D. J.; Newell, A. C., J. Math. Phys., 19, 798 (1978)
[7] Svinolupov, S. I.; Yamilov, R. I., Phys. Lett. A, 160, 548 (1991)
[8] Svinolupov, S. I.; Yamilov, R. I., Teoret. Mat. Fiz.. Teoret. Mat. Fiz., Theor. Math. Phys., 98, 2, 139 (1994), English Transl. in:
[9] Loos, O., (Lecture Notes in Math., Vol. 480 (1975), Springer: Springer Berlin), Jordan Pairs · Zbl 0301.17003
[10] Neher, E., (Lecture Notes in Math., Vol. 1280 (1987), Springer: Springer Berlin)
[11] Meyberg, K., Math. Z. B, 115, 58 (1970)
[12] Fordy, A. P.; Kulish, P. P., Commun. Math. Phys., 89, 427 (1983)
[13] Athorn, C.; Fordy, A. P., J. Math. Phys., 28, 2018 (1987)
[14] Fordy, A. P., J. Phys. A, 17, 1235 (1984)
[15] Fokas, A. S.; Fuchssteiner, B., Phys. Lett. A, 86, 341 (1981)
[16] Fuchssteiner, B., Progr. Theor. Phys., 70, 1508 (1983)
[17] Fokas, A. S., Stud. Appl. Math., 77, 253 (1987)
[18] Calogero, F.; Degasperis, A., (Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, I (1982), North-Holland: North-Holland Amsterdam) · Zbl 0501.35072
[19] Bruschi, M.; Levi, D.; Ragnisco, O., Nuovo Cimento A, 48, 213 (1978)
[20] Oevel, W.; Zhang, H.; Fuchssteiner, B., Progr. Theor. Phys., 81, 294 (1989)
[21] Strampp, W.; Oevel, W., Progr. Theor. Phys., 74, 922 (1985)
[22] Yamilov, R. I., Classification of Toda type scalar lattices, (Makhankov, V.; Puzynin, I.; Pashaev, O., Proc. Workshop on Nonlinear Evolution Equations and Dynamical Systems (1993), World Scientific: World Scientific Singapore), 423
[23] Cherdantsev, I.; Yamilov, R., (CRM Proc. Lecture Notes, 9 (1996)), 51
[24] Svinolupov, S. I.; Sokolov, V. V., Teoret. Mat. Fiz., 108, 388 (1996)
[25] Adler, V. E., Phys. Lett. A, 190, 53 (1994)
[26] F. Calogero, Tricks of the trade: relating and deriving solvable and integrable dynamical systems, to be published.; F. Calogero, Tricks of the trade: relating and deriving solvable and integrable dynamical systems, to be published.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.