The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schrödinger, and Hamilton--Jacobi--Bellman Equations
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914507366858752 |
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| author | Covei, Dragos-Patru |
| author_facet | Covei, Dragos-Patru |
| contents | This study develops a unified mathematical framework for the analysis of radial differential equations, revealing a fundamental connection between three distinct classes of problems: the nonlinear Riccati equation, the linear Schrödinger equation, and the Hamilton--Jacobi--Bellman equation for stochastic control. We establish the existence and uniqueness of regular solutions on both bounded and unbounded domains, deriving sharp growth rates and exact asymptotic plateaus through a general barrier theory. A detailed sensitivity analysis of the noise intensity parameter identifies the transition between deterministic and diffusion-dominated regimes via singular perturbation methods. These theoretical results are reinforced by numerical simulations that validate the predicted feedback laws, confirm the convexity--concavity structure of the triality, and illustrate the stability of the system. The resulting framework clarifies the duality between global wave functions and local dynamical drifts, providing a rigorous foundation for the study of multidimensional stochastic processes under central potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27772 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schrödinger, and Hamilton--Jacobi--Bellman Equations Covei, Dragos-Patru Analysis of PDEs This study develops a unified mathematical framework for the analysis of radial differential equations, revealing a fundamental connection between three distinct classes of problems: the nonlinear Riccati equation, the linear Schrödinger equation, and the Hamilton--Jacobi--Bellman equation for stochastic control. We establish the existence and uniqueness of regular solutions on both bounded and unbounded domains, deriving sharp growth rates and exact asymptotic plateaus through a general barrier theory. A detailed sensitivity analysis of the noise intensity parameter identifies the transition between deterministic and diffusion-dominated regimes via singular perturbation methods. These theoretical results are reinforced by numerical simulations that validate the predicted feedback laws, confirm the convexity--concavity structure of the triality, and illustrate the stability of the system. The resulting framework clarifies the duality between global wave functions and local dynamical drifts, providing a rigorous foundation for the study of multidimensional stochastic processes under central potentials. |
| title | The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schrödinger, and Hamilton--Jacobi--Bellman Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.27772 |