International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 10, Pages 587-614
On the compatible weakly nonlocal Poisson brackets of hydrodynamic type
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.
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 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 and are nondegenerate.