Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley  type Systems

Partha Guha

Contact the author: Please use for correspondence this email.
Submission date: 13. Sep. 2007
Pages: 35
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
Bibtex
MSC-Numbers: 37K65, 37K10, 58D05
Keywords and phrases: Virasoro algebra, Sobolev metric, Euler-Poincare
Download full preprint: PDF (295 kB)

Abstract:
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.