A. Comech, S. Cuccagna and D. Pelinovsky

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

SIAM Journal on Mathematical Analysis 39, 1-33 (2007)

Abstract:
We prove the instability of a ``critical'' soliton of the generalized Korteweg - de Vries equation, the one that is at the border of the stability region (solitons of higher speeds are stable and of lower speed are unstable). 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.

Keywords:
KORTEWEG-DE VRIES EQUATION, HAMILTONIAN SYSTEMS, INSTABILITY OF NONLINEAR WAVES, MULTIPLE EIGENVALUES, NORMAL FORMS, EXPONENTIALLY WEIGHTED SPACES