| Abstract: |
| We study the bi-Hamiltonian property for the hierarchy of a 3-component Degasperis-Procesi (3-DP) equation. We show that Hamiltonian functionals of the hierarchy in negative direction are local, and in both directions are homogenous. Two different Liouville transformations for the equation are constructed via a reciprocal transformation. The first transformation shows that the associated equation is a reduced negative modified Yajima-Oikawa system. The second transformation shows that the associated system is a reduced negative generalized mKdV system and passes the Painlev\`{e} test, besides Hamiltonian structures of the 3-DP equation under this Liouville transformation are also discussed. Moreover, we consider a limit for the 3-DP equation. |
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