Asymptotic Stability for KdV Solitons in Weighted Spaces via Iteration

In this paper, we reconsider the well-known result of Pego-Weinstein \cite{MR1289328} that soliton solutions to the Korteweg-deVries 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 \cite{PR}. This will be possible because the exponential approach rate obtained here will defeat the polynomial loss in traditional applications of the $I$-method \cite{MR1995945}, \cite{MR1951312}, \cite{pigottorb}.