×

Integrable systems generated by a constant solution of the Yang-Baxter equation. (English. Russian original) Zbl 0895.35097

Funct. Anal. Appl. 30, No. 4, 275-277 (1996); translation from Funkts. Anal. Prilozh. 30, No. 4, 68-71 (1996).
A class of integrable systems of the form \(q_t= r\cdot q_{xy} +F(q,q_y)\) is constructed. Here \(q(x,y,t)\) is a vector and \(r:{\mathfrak g} \to{\mathfrak g}\) is a constant linear operator on the Lie algebra \({\mathfrak g}\) that satisfies the Yang-Baxter equation \[ r \biggl(\bigl[ r(a),b\bigr] -\bigl[r(b), a\bigr]\biggr) -\bigl[r(a), r(b)\bigr] =0 \] for any \(a,b\in{\mathfrak g}\). The considered equations have local higher symmetries and conservation laws. They are similar to the \(N\)-wave equation.
Reviewer: A.Rudy (Minsk)

MSC:

35Q58 Other completely integrable PDE (MSC2000)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B99 Lie algebras and Lie superalgebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] V. G. Drinfeld and V. V. Sokolov, ”Lie algebras and equations of KdV type,” in: Sovremennye Problemy Matematiki. Itogi Nauki i Tekhniki, Vol. 24, VINITI, Moscow, 1984, pp. 81–180.
[2] V. G. Drinfeld, Dokl. Akad. Nauk SSSR,273, No. 3, 531–534 (1983).
[3] M. A. Semenov-Tian-Shansky, Funkts. Anal. Prilozhen.,17, No. 4, 17–33 (1983).
[4] A. Medina, C. R. Acad. Sci. Paris, Ser. A.,286, 173–177 (1978).
[5] S. I. Svinolupov, Phys. Lett. A.,135, 32–36 (1987).
[6] I. Z. Golubchik, V. V. Sokolov, and S. I. Svinolupov, A New Class of Nonassociative Algebras and a Generalized Factorization Method, Preprint ESI,53, Wien, Austria, 1993.
[7] A. G. Reyman and M. A. Semenov-Tian-Shansky, Funkts. Anal. Prilozhen.,14, No. 2, 77–78 (1980).
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.