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Theoretical and Mathematical Physics

, Volume 167, Issue 1, pp 403–420 | Cite as

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

  • O. I. MokhovEmail author
Article

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.

Keywords

bi-Hamiltonian system of hydrodynamic type Riemann invariant compatible metrics diagonalizable affinor bi-Hamiltonian structure bi-Hamiltonian affinor integrable system 

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Copyright information

© MAIK/Nauka 2011

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRASChernogolovka, Moscow OblastRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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