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.

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