  The influence of the intramolecular anharmonicity and the strong
vibron-phonon coupling on the two-vibron dynamics in an $\alpha$-helix protein
is studied within a modified Davydov model. The intramolecular anharmonicity of
each amide-I vibration is considered and the vibron dynamics is described
according to the small polaron approach. A unitary transformation is performed
to remove the intramolecular anharmonicity and a modified Lang-Firsov
transformation is applied to renormalize the vibron-phonon interaction. Then, a
mean field procedure is realized to obtain the dressed anharmonic vibron
Hamiltonian. It is shown that the anharmonicity modifies the vibron-phonon
interaction which results in an enhancement of the dressing effect. In
addition, both the anharmonicity and the dressing favor the occurrence of two
different bound states which the properties strongly depend on the interplay
between the anharmonicity and the dressing. Such a dependence was summarized in
a phase diagram which characterizes the number and the nature of the bound
states as a function of the relevant parameters of the problem. For a
significant anharmonicity, the low frequency bound states describe two vibrons
trapped onto the same amide-I vibration whereas the high frequency bound states
refer to the trapping of the two vibrons onto nearest neighbor amide-I
vibrations.
