The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schr\"{o}dinger, and Hamilton--Jacobi--Bellman Equations

Abstract

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 Schr\"odinger 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.