A Paley-Wiener Theorem for Periodic Scattering with Applications to the Korteweg-de Vries Equation

TL;DR

This paper establishes a Paley-Wiener type theorem for periodic scattering problems, characterizing when potential differences have finite support, and applies it to unique continuation results for the Korteweg-de Vries (KdV) and modified KdV equations.

Contribution

It provides necessary and sufficient conditions on reflection coefficients for finite support of potential differences and applies these to unique continuation properties of KdV solutions.

Findings

Characterization of finite support potentials via reflection coefficients

Unique continuation results for KdV solutions

Extension to modified KdV via Miura transform

Abstract

Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.