A complex logistic equation for universal energy evolution in hadronic elastic scattering

We introduce a universal evolution equation for elastic scattering of hadrons, derived from Regge field theory (RFT) and solved in closed analytical form. The equation emerges from a complex logistic structure and evolves initial amplitude profiles taken from existing models at fixed energy, reproducing both the differential cross sections and the integrated quantities in a broad energy range. We prove that it admits a unique solution for each initial condition and rigorously satisfies unitarity, the Froissart-Martin bound, and dispersion relations. The dynamics are governed by two physically meaningful parameters: the effective Pomeron mass $\epsilon_{\mathcal{P}}$ and the nonlinear coupling $\lambda$, both fitted at a single energy. Our approach offers a minimal, yet predictive framework for saturation and unitarization in elastic scattering and may provide a useful bridge toward small-$x$ QCD evolution.