Compatible metrics and the diagonalizability of nonlocally biHamiltonian systems of hydrodynamic type
Abstract
We study biHamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal biHamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the biHamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally biHamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differentialgeometric 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 biHamiltonian 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 biHamiltonian structures; we briefly review the necessary notions and results in those two theories.
 Publication:

Theoretical and Mathematical Physics
 Pub Date:
 April 2011
 DOI:
 10.1007/s112320110032z
 Bibcode:
 2011TMP...167..403M
 Keywords:

 biHamiltonian system of hydrodynamic type;
 Riemann invariant;
 compatible metrics;
 diagonalizable affinor;
 biHamiltonian structure;
 biHamiltonian affinor;
 integrable system