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:
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