Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation

Comech, Andrew; Cuccagna, Scipio; Pelinovsky, Dmitry

Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation

Authors

Comech, Andrew

Cuccagna, Scipio

Pelinovsky, Dmitry

Description

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.
34 pages, 3 figures