  This paper sheds new light on the stability properties of solitary wave
solutions associated with models of Korteweg-de Vries and
Benjamin\&Bona\&Mahoney type, when the dispersion is very lower. Via an
approach of compactness, analyticity and asymptotic perturbation theory, we
establish sufficient conditions for the existence of exponentially growing
solutions to the linearized problem and so a criterium of linear instability of
solitary waves is obtained for both models. Moreover, the nonlinear stability
and linear instability of the ground states solutions for both models is
obtained for some specific regimen of parameters. Via a Lyapunov strategy and a
variational analysis we obtain the stability of the blow-up of solitary waves
for the critical fractional KdV equation.
  The arguments presented in this investigation has prospects for the study of
the instability of traveling waves solutions of other nonlinear evolution
equations.
