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

Partha Guha


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.

Sep 13, 2007
Sep 13, 2007
MSC Codes:
37K65, 37K10, 58D05
Virasoro algebra, Sobolev metric, Euler-Poincare

Related publications

2007 Journal Open Access
Partha Guha

Euler-Poincaré formalism of (two component) Degasperis-Procesi and Holm-Staley type systems

In: Journal of nonlinear mathematical physics, 14 (2007) 3, pp. 390-421