The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schrödinger, and Hamilton--Jacobi--Bellman Equations

Abstract: This study establishes a comprehensive mathematical framework for the analysis of radial differential equations, identifying 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 prove the existence and uniqueness of regular solutions on both bounded and unbounded domains, identifying sharp growth rates and asymptotic behaviors. Furthermore, we conduct a sensitivity analysis of the noise intensity parameter, characterizing the transitions between deterministic and diffusion-dominated regimes through the lens of singular perturbation theory. Our theoretical results are complemented by numerical simulations that validate the predicted feedback laws and the structural stability of the system. This unified approach provides deep insights into the duality between global wave functions and local dynamical drifts, offering a rigorous basis for analyzing multidimensional stochastic processes under central potentials.