A two-component generalization of the Degasperis–Procesi equation

Abstract

We present two different hamiltonian extensions of the Degasperis -Procesi equation to the two component equations. The construction based on the observation that the second Hamiltonian operator of the Degasperis -Procesi equation could be considered as the Dirac reduced Poisson tensor of the second Hamiltonian operator of the Boussinesq equation. The first extension is generated by the Hamiltonian operator which is a Dirac reduced operator of the generalized but degenerated second Hamiltonian operator of the Boussinesq equation. The second one is obtained by the N = 2 supersymmetric extension of the mentioned method. As the byproduct of this procedure we obtained the Hamiltonian system of interacting equations which contains the Camassa -Holm and Degasperis -Procesi equation.

A Synthetical Two‐Component Model with Peakon Solutions

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Perturbational self-similar solutions for the 2-component Degasperis--Procesi system via a characteristic method

“…In [33], Popowicz constructed the Hamiltonian structures for certain parameters. In [15], Lin and Guo analyzed some aspects of blowup mechanism, traveling wave solutions, and the persistence properties of the system.…”

Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons

“…31. However, systems ͑1.3͒-͑1.6͒ do not have the peakon solitons in the form of a superposition of multipeakons, and system ͑1.7͒ does not have H 1 weak solution.…”