Schouten tensor and bi-Hamiltonian systems of hydrodynamic type. (English) Zbl 1111.37051
Summary: Necessary conditions are derived for the existence of a bi-Hamiltonian structure for a given hydrodynamic type system. One of the conditions is the vanishing of its Haantjes tensor. A theorem is proved on the canonical forms of the generic bi-Hamiltonian systems. The Schouten \((2,1)\)-tensor \(S_i^{jk}\) is connected with any Hamiltonian system of hydrodynamic type. The complete symmetry of the \((3,0)\)-tensor \(S^{ijk}\) is demonstrated. Necessary conditions for the existence of a single nondegenerate Hamiltonian structure are obtained in terms of the special differential \(k\)-forms \(\Omega_{p_1\cdots p_k} (u_1,\dots,u_k)\).
MSC:
37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
35Q53 | KdV equations (Korteweg-de Vries equations) |
37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
53C80 | Applications of global differential geometry to the sciences |
References:
[1] | Schouten J. A., Convegno Internazionale di Geometria Differenziale pp 1– (1954) |
[2] | DOI: 10.1063/1.523777 · Zbl 0383.35065 · doi:10.1063/1.523777 |
[3] | P. J. Olver, in BiHamiltonian Systems, Pitman Research Notes in Mathematics Series 157, edited by B. D. Sleeman and R. J. Jarvis (Longman Scientific and Technical, New York, 1987), pp. 176–193. |
[4] | DOI: 10.1016/0375-9601(90)90775-J · doi:10.1016/0375-9601(90)90775-J |
[5] | Kosmann-Schwarzbach Y., Ann. Inst. Henri Poincare, Sect. A 53 pp 35– (1990) |
[6] | DOI: 10.1007/BF01466596 · Zbl 0602.58017 · doi:10.1007/BF01466596 |
[7] | DOI: 10.1007/BF02099623 · Zbl 0861.58021 · doi:10.1007/BF02099623 |
[8] | DOI: 10.1070/RM1989v044n06ABEH002300 · Zbl 0712.58032 · doi:10.1070/RM1989v044n06ABEH002300 |
[9] | DOI: 10.1070/RM1990v045n03ABEH002351 · Zbl 0712.35080 · doi:10.1070/RM1990v045n03ABEH002351 |
[10] | DOI: 10.1016/0375-9601(92)90365-S · doi:10.1016/0375-9601(92)90365-S |
[11] | Nijenhuis A., Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci. 54 pp 200– (1951) |
[12] | Haantjes J., Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci. 58 pp 158– (1955) |
[13] | Tsarev S. P., Dokl. Akad. Nauk SSSR 282 pp 534– (1985) |
[14] | DOI: 10.1007/BF02517890 · Zbl 0872.58026 · doi:10.1007/BF02517890 |
[15] | DOI: 10.1007/978-0-387-21792-5 · doi:10.1007/978-0-387-21792-5 |
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.