Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley type Systems
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Submission date: 13. Sep. 2007
published in: Journal of nonlinear mathematical physics, 14 (2007) 3, p. 390-421
DOI number (of the published article): 10.2991/jnmp.2007.14.3.8
MSC-Numbers: 37K65, 37K10, 58D05
Keywords and phrases: Virasoro algebra, Sobolev metric, Euler-Poincare
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In this paper we propose an Euler-Poincaré formalism of
the Degasperis and Procesi (DP) equation. This is a second member of a
one-parameter family of partial differential equations,
known as b-field equations. This one-parameter family of pdes
includes the integrable Camassa-Holm equation as a first member.
We show that our Euler-Poincaré formalism exactly coincides
with the Degasperis-Holm-Hone (DHH) Hamiltonian framework. We obtain the
DHH Hamiltonian structues of the DP equation from our method.
Recently this new equation has been generalized by Holm and Staley by adding
viscosity term. We also discuss
Euler-Poincaré formalism of the Holm-Staley equation. In the
second half of the paper we consider a generalization of the
Degasperis and Procesi (DP) equation with two dependent variables.
we study the Euler-Poincaré framework of
the 2-component Degasperis-Procesi equation. We also mention about
the b-family equation.