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 ij (u) and g ij (u), the affinor υ j i (u) = g 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.