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