Geodesic
Compatible metrics and the~diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type
Teoretičeskaâ i matematičeskaâ fizika, Tome 167 (2011) no. 1, pp. 3-22

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We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) $V^i_j(u)$ of this system of hydrodynamic type, the metrics $g^{ij}_1(u)$ and $g^{ij}_2(u)$, the affinor $v^i_j(u)=g_1^{is}(u)g_{2,sj}(u)$, and also the affinors $(w_{1,n})^i_j(u)$ and $(w_{2,n})^i_j(u)$ of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.
Keywords: bi-Hamiltonian system of hydrodynamic type, compatible metrics, diagonalizable affinor, bi-Hamiltonian structure, bi-Hamiltonian affinor, integrable system.
Mots-clés : Riemann invariant 
@article{TMF_2011_167_1_a0,
     author = {O. I. Mokhov},
     title = {Compatible metrics and the~diagonalizability of nonlocally {bi-Hamiltonian} systems of hydrodynamic type},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--22},
     publisher = {mathdoc},
     volume = {167},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2011_167_1_a0/}
}
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O. I. Mokhov. Compatible metrics and the~diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type. Teoretičeskaâ i matematičeskaâ fizika, Tome 167 (2011) no. 1, pp. 3-22. http://geodesic.mathdoc.fr/item/TMF_2011_167_1_a0/