Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type
Abstract
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{j/i}(u) of this system of hydrodynamic type, the metrics g{/1 ij}(u) and g{/2 ij}(u), the affinor υ{j/i}(u) = g{/1 is}(u)g 2,sj(u), and also the affinors (w 1,n){j/i}(u) and (w 2,n){j/i}(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.
- Publication:
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Theoretical and Mathematical Physics
- Pub Date:
- April 2011
- DOI:
- 10.1007/s11232-011-0032-z
- Bibcode:
- 2011TMP...167..403M
- Keywords:
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- bi-Hamiltonian system of hydrodynamic type;
- Riemann invariant;
- compatible metrics;
- diagonalizable affinor;
- bi-Hamiltonian structure;
- bi-Hamiltonian affinor;
- integrable system