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

Stability properties of solitary waves for fractional KdV and BBM equations