On compatible metrics and diagonalizability of non-locally bi-Hamiltonian systems of hydrodynamic type

TL;DR

This paper proves that non-singular non-local bi-Hamiltonian systems of hydrodynamic type are diagonalizable and can be expressed in local coordinates where all related structures are diagonal, extending the theory of compatible metrics.

Contribution

It establishes the diagonalizability of non-singular non-local bi-Hamiltonian hydrodynamic systems and existence of local coordinates simplifying their structures.

Findings

Systems are diagonalizable in suitable local coordinates.

All related matrix and metric objects are diagonal in these coordinates.

The results extend the theory of compatible metrics and bi-Hamiltonian structures.

Abstract

We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all the related matrix differential-geometric objects, namely, the matrix V^i_j(u) of this system of hydrodynamic type, the metrics g^{ij}_1(u) and g^{ij}_2(u) and the affinors (w_{1, n})^i_j(u) and (w_{2,n})^i_j(u) of the non-singular non-local bi-Hamiltonian structure of this system, are diagonal in these local coordinates. The proof is a natural consequence of the general results of the theory of compatible metrics and the theory of non-local bi-Hamiltonian structures developed earlier by the present author.