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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.19129 |
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| _version_ | 1866912451247734784 |
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| author | He, Yuchao Wu, Mengda Xia, Yonghui Zhang, Meirong |
| author_facet | He, Yuchao Wu, Mengda Xia, Yonghui Zhang, Meirong |
| contents | 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 $\|\hat q\|_{Lp}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A novel and application-oriented inverse nodal problem for Sturm-Liouville operators He, Yuchao Wu, Mengda Xia, Yonghui Zhang, Meirong Classical Analysis and ODEs 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 $\|\hat q\|_{Lp}$ 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. |
| title | A novel and application-oriented inverse nodal problem for Sturm-Liouville operators |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2505.19129 |