A novel and application-oriented inverse nodal problem for Sturm-Liouville operators

TL;DR

This paper introduces a new optimization-based framework for solving inverse nodal problems in Sturm-Liouville operators, with applications in seismic and sonar detection, providing explicit reconstruction methods and uniqueness results.

Contribution

It develops a systematic approach to reconstruct potentials from nodal data, reformulates the problem via nonlinear Schrödinger equations, and proves uniqueness under certain conditions.

Findings

Potential $ ilde q$ can be explicitly reconstructed from nodal data.

When $q_0$ is constant, the Schrödinger equations are integrable and $ ilde q$ is periodic.

Uniqueness of the potential $ ilde q$ is established for $p > 3/2$.

Abstract

This paper develops a methodological framework for addressing a novel and application-oriented inverse nodal problem in Sturm-Liouville operators, having significant applications in seismic wave analysis and submarine underwater radar (sonar) detection. By utilizing a given finite set of nodal data, we propose an optimization framework to find the potential $\hat q$ that is most closely approximating a predefined target potential $q_0$. The inverse nodal optimization problem is reformulated as a solvability problem for a class of nonlinear Schr\"odinger equations, enabling systematic investigation of the inverse nodal problem. {As an example, when the constant target potential $q_0$ is considered, we find that the Schr\"odinger equations are completely integrable and conclude that the potential $\hat q$ is `periodic' in a certain sense. Furthermore, the reconstruction of $\hat q$ is reduced to solving a system of three featured parameters, thereby establishing an explicit quantitative relationship between $\|\hat q\|_{Lp}$ and $T_*$. Of importance, we prove the uniqueness of the potential $\hat q$ when $p>3/2$. These new findings represent a substantial advancement in this field of study. Our methodology also bridges theoretical rigor with practical applicability, addressing scenarios where only partial nodal information is available.