Inverse dynamic problem for the wave equation with periodic boundary conditions

TL;DR

This paper addresses the inverse dynamic problem for the wave equation with a potential on a periodic interval, utilizing boundary triplets and response operators to establish relationships between dynamic and spectral data.

Contribution

It introduces a novel approach using boundary triplets and response operators to solve the inverse problem for the wave equation with periodic boundary conditions.

Findings

Derived equations for the inverse problem

Established relationships between dynamic and spectral inverse data

Provided a framework for reconstructing the potential

Abstract

We consider the inverse dynamic problem for the wave equation with a potential on an interval $(0,2\pi)$ with periodic boundary conditions. We use a boundary triplet to set up the initial-boundary value problem. As an inverse data we use a response operator (dynamic Dirichlet-to-Neumann map). Using the auxiliary problem on the whole line, we derive equations of the inverse problem. We also establish the relationships between dynamic and spectral inverse data.