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We prove the instability of a ``critical'' solitary wave of the generalizedKorteweg -- de Vries equation, the one with the speed at the border between thestability and instability regions. The instability mechanism involved is``purely nonlinear'', in the sense that the linearization at a critical solitondoes not have eigenvalues with positive real part. We prove that criticalsolitons correspond generally to the saddle-node bifurcation of two branches ofsolitons.

Nonlinear Instability of a Critical Traveling Wave in the Generalized Korteweg–de Vries Equation