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 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 \(\hat{q}\) 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 \(\Vert \hat{q}\Vert _{\mathcal {L}^p}\) 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.
