Jordan algebras and generalized Korteweg-de Vries equations. (English. Russian original) Zbl 0746.35044
Theor. Math. Phys. 87, No. 3, 611-620 (1991); translation from Teor. Mat. Fiz. 87, No. 3, 391-403 (1991).
Summary: Integrability criteria for many-field Korteweg-de Vries equations are obtained. A one-to-one correspondence between such equations and Jordan algebras is established. It is shown that the so-called irreducible systems correspond to simple Jordan algebras.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 17C20 | Simple, semisimple Jordan algebras |
| 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 |
References:
| [1] | C. Athorne and A. Fordy, J. Phys. A,20, 1377 (1987). · Zbl 0642.58034 · doi:10.1088/0305-4470/20/6/021 |
| [2] | V. G. Drinfel’d and V. V. Sokolov, in: Modern Problems of Mathematics, Vol. 24 [in Russian], VINITI, Moscow (1984), pp. 81-181. |
| [3] | N. Jacobson, Structure and Representations of Jordan Algebras, Providence (1968). · Zbl 0218.17010 |
| [4] | P. Jordan, J. von Neumann, and E. Wigner, Ann. Math.,36, 29 (1934). · Zbl 0008.42103 · doi:10.2307/1968117 |
| [5] | A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Nearly Associative Rings [in Russian], Nauka, Moscow (1978). · Zbl 0445.17001 |
| [6] | A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Usp. Mat. Nauk,42, 3 (1987). |
| [7] | A. V. Mikhailov, V. V. Sokolov, and A. B. Shabat, ?The symmetry approach to the classification of integrable equations,? in: Integrability and Kinetic Equations for Solitons [in Russian], Naukova Dumka, Kiev (1989). |
| [8] | P. Olver, Applications of Lie Groups to Differential Equations [Russian translation], Mir, Moscow (1989). · Zbl 0743.58003 |
| [9] | A. Albert, Ann. Math.,48, 546 (1947). · Zbl 0029.01003 · doi:10.2307/1969128 |
| [10] | A. Albert, Trans. Am. Math. Soc.,69, 503 (1950). |
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.
