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