Asymptotic stability for KdV solitons in weighted spaces via iteration

Abstract

In this paper, we reconsider the well-known result of Pego–Weinstein ( Comm. Math. Phys. 2 (1994) 305–349) that soliton solutions to the Korteweg–de Vries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result in the setting of modern well-posedness function spaces. We obtain asymptotic stability in the exponentially weighted space via an iteration argument. Our purpose here is to lay the groundwork to use the $I$-method to obtain asymptotic stability below $H^{1}$, which will be done in a second, forthcoming paper (Asymptotic stability for KdV solitons in weighted $H^{s}$ spaces. Preprint). This will be possible because the exponential approach rate obtained here will defeat the polynomial loss in traditional applications of the $I$-method ( Commun. Pure Appl. Anal. 2 (2003) 277–296, Discrete Contin. Dyn. Syst. 9 (2003) 31–54, Commun. Pure Appl. Anal. 13 (2014) 389–418).