  In this paper we establish a rigorous spectral stability analysis for
solitary waves associated to a generalized fractional Benjamin-Bona-Mahony type
equation. Besides the well known smooth and positive solitary wave with large
wave speed, we present the existence of smooth negative solitary waves having
small wave speed. The spectral stability is then determined by analysing the
behaviour of the associated linearized operator around the wave restricted to
the orthogonal of the tangent space related to the momentum at the solitary
wave. Since the analytical solution is not known, we generate the negative
solitary waves numerically by using Petviashvili method. We also present some
numerical experiments to observe the stability properties of solitary waves for
various values of the order of nonlinearity and fractional derivative. Some
remarks concerning the orbital stability are also celebrated.

Analysis of PDEs