2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116330We prove the instability of a ``critical'' solitary wave of the generalized Korteweg -- de Vries equation, the one with the speed at the border between the stability and instability regions. The instability mechanism involved is ``purely nonlinear'', in the sense that the linearization at a critical soliton does not have eigenvalues with positive real part. We prove that critical solitons correspond generally to the saddle-node bifurcation of two branches of solitons.34 pages, 3 figuresAnalysis of PDEsMathematical Physicstext