arXiv subjects
Publications and source records attributed to Allison Byars.
We prove a modified scattering and asymptotic completeness for the derivative nonlinear Schr\"odinger equation. This is the first result proving asymptotic completeness in a quasilinear setting. Our approach combines the method of testing by wave packets, introduced by Ifrim and Tataru, a bootstrap argument, and the Klainerman Sobolev vector field method.
In this paper, we consider the derivative nonlinear Schr\"odinger (DNLS) equation. While the existence theory has been intensely studied, properties like dispersive estimates for the solutions have not yet been investigated. Here we address this question for the problem with small and localized data, and show that a dispersive estimate for the solution holds globally in time. For the proof of our result we use vector field methods combined with the \emph{testing by wave packets method}, whose implementation in this problem is novel.
Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere (with respect to their respective invariant measures) intersection.