The aim of this paper is to provide a proof of the (conditional) orbital stability of solitary waves solutions to the fractional Korteweg-de Vries equation (fKdV) and to the fractional Benjamin-Bona-Mahony (fBBM) equation in the $L^2$ subcritical case. We also discuss instability and its possible scenarios.
"Remarks on the orbital stability of ground state solutions of fKdV and related equations." Adv. Differential Equations 20 (9/10) 835 - 858, September/October 2015. https://doi.org/10.57262/ade/1435064515