Asymptotic Stability for Kdv Solitons in Weighted Spaces via Iteration

In this paper, we reconsider the well-known result of PegoWeinstein [17] 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 H1, which will be done in a second, forthcoming paper [19]. This will be possible because the exponential approach rate obtained here will defeat the polynomial loss in traditional applications of the I-method [20], [6], [18].