Nonlinear instability of a critical traveling wave in the generalized   Korteweg -- de Vries equation
  We 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. 
Andrew Comech, Scipio Cuccagna, and Dmitry Pelinovsky