A two-component generalization of the Degasperis–Procesi equation

Abstract

We present two different hamiltonian extensions of the Degasperis - Procesi\nequation to the two component equations. The construction based on the\nobservation that the second Hamiltonian operator of the Degasperis - Procesi\nequation could be considered as the Dirac reduced Poisson tensor of the second\nHamiltonian operator of the Boussinesq equation. The first extension is\ngenerated by the Hamiltonian operator which is a Dirac reduced operator of the\ngeneralized but degenerated second Hamiltonian operator of the Boussinesq\nequation. The second one is obtained by the N=2 supersymmetric extension of the\nmentioned method. As the byproduct of this procedure we obatined the\nHamiltonian system of interacting equations which contains the Camassa - Holm\nand Degasperis - Procesi equation.\n