Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation
Andrew Komech, Scipio Cuccagna, and Dmitry E. Pelinovsky
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Submission date: 26. Jul. 2006
published in: SIAM journal on mathematical analysis, 39 (2007) 1, p. 1-33
DOI number (of the published article): 10.1137/06065150
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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.