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.
Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation / Cuccagna, Scipio; Andrew, Comech; Dmitry, Pelinovsky. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - STAMPA. - 39:1(2007), pp. 1-33. [10.1137/060651501]
Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation
CUCCAGNA, Scipio;
2007
Abstract
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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