International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 10, Pages 587-614
doi:10.1155/S0161171202202069

On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

Andrei Ya. Maltsev1,2

1L. D. Landau Institute for Theoretical Physics, Kosygina Street 2, Moscow 117940, Russia
2University of Maryland, College Park 20742, MD, USA

Received 5 February 2002

Copyright © 2002 Andrei Ya. Maltsev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo-Riemannian metric g(0) νμ and also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo-Riemannian metrics g(0) νμ and g(1) νμ are nondegenerate.