40 research outputs found
Non-Homogeneous Hydrodynamic Systems and Quasi-St\"ackel Hamiltonians
In this paper we present a novel construction of non-homogeneous hydrodynamic
equations from what we call quasi-St\"ackel systems, that is non-commutatively
integrable systems constructed from appropriate maximally superintegrable
St\"ackel systems. We describe the relations between Poisson algebras generated
by quasi-St\"ackel Hamiltonians and the corresponding Lie algebras of vector
fields of non-homogeneous hydrodynamic systems. We also apply St\"ackel
transform to obtain new non-homogeneous equations of considered type
St\"ackel transform of Lax equations
We construct Lax pairs for a wide class of St\"ackel systems by applying the
multi-parameter St\"ackel transform to Lax pairs of a suitably chosen systems
from the seed class. For a given St\"ackel system, the obtained set of
non-equivalent Lax pairs is parametrized by an arbitrary function
Invertible coupled KdV and coupled Harry Dym hierarchies
In this paper we discuss the conditions under which the coupled KdV and
coupled Harry Dym hierarchies possess inverse (negative) parts. We further
investigate the structure of nonlocal parts of tensor invariants of these
hierarchies, in particular, the nonlocal terms of vector fields, conserved
one-forms, recursion operators, Poisson and symplectic operators. We show that
the invertible cKdV hierarchies possess Poisson structures that are at most
weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures
with nonlocalities of the third order