Asymptotic stability of solitons of the subcritical gKdV equations revisited

Abstract

We consider the generalized Korteweg–de Vries (gKdV) equations in the subcritical cases p = 2, 3 or 4. The first objective of this paper is to present a direct, simplified proof of the asymptotic stability of solitons in the energy space H1, which was proved by the same authors in Martel and Merle (2001 Arch. Ration. Mech. Anal. 157 219–54).Then, in the case of the KdV equation, we show the optimality of the result by constructing a solution which behaves asymptotically as a soliton located on a curve which is a logarithmic perturbation of a line. This example justifies the apparently weak control on the location of the soliton in the asymptotic stability result.