2 research outputs found
On a Camassa-Holm type equation with two dependent variables
We consider a generalization of the Camassa Holm (CH) equation with two
dependent variables, called CH2, introduced by Liu and Zhang. We briefly
provide an alternative derivation of it based on the theory of Hamiltonian
structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the
same algebra underlying the NLS hierarchy. We study the structural properties
of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and
provide its Lax representation. Then we explicitly discuss how to construct
classes of solutions, both of peakon and of algebro-geometrical type. We
finally sketch the construction of a class of singular solutions, defined by
setting to zero one of the two dependent variables.Comment: 22 pages, 2 figures. A few typos correcte
Gaudin Models and Bending Flows: a Geometrical Point of View
In this paper we discuss the bihamiltonian formulation of the (rational XXX)
Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued
spins. In particular, we focus on the homogeneous models. We find a pencil of
Poisson brackets that recursively define a complete set of integrals of the
motion, alternative to the set of integrals associated with the 'standard' Lax
representation of the Gaudin model. These integrals, in the case of su(2),
coincide wih the Hamiltonians of the 'bending flows' in the moduli space of
polygons in Euclidean space introduced by Kapovich and Millson. We finally
address the problem of separability of these flows and explicitly find
separation coordinates and separation relations for the r=2 case.Comment: 27 pages, LaTeX with amsmath and amssym