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

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 $\widehat{q}$ 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 Schrödinger 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 Schrödinger equations are completely integrable and conclude that the potential $\widehat{q}$ is `periodic’ in a certain sense. Furthermore, the reconstruction of $\widehat{q}$ is reduced to solving a system of three featured parameters, thereby establishing an explicit quantitative relationship between $\mathrm{\parallel }\widehat{q}{\mathrm{\parallel }}_{Lp}$ and $T_{\ast }$. Of importance, we prove the uniqueness of the potential $\widehat{q}$ when $p>3\mathrm{/}2$. 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.