On a Camassa-Holm type equation with two dependent variables

Details On a Camassa-Holm type equation with two dependent variables On a Camassa-Holm type equation with two dependent variables

2005-05-27

arxiv.org/abs/nlin/0505059v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

22 pages, 2 figures. A few typos corrected

Scientific paper

10.1088/0305-4470/39/2/004

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.

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