  The generalized fractional Benjamin-Bona-Mahony (gfBBM) equation models the
propagation of small amplitude long unidirectional waves in a nonlocally and
nonlinearly elastic medium. The equation involves two fractional terms unlike
the well-known fBBM equation. In this paper, we prove local existence and
uniqueness of the solutions for the Cauchy problem. The sufficient conditions
for the existence of solitary wave solutions are obtained. The Petviashvili
method is proposed for the generation of the solitary wave solutions and their
evolution in time is investigated numerically by Fourier spectral method. The
efficiency of the numerical methods is tested and the relation between
nonlinearity and fractional dispersion is observed by various numerical
experiments.
