This work presents a comprehensive variational and numerical analysis of hyperbolic-function optical beams—hyperbolic tangent-Gaussian (ThG), sine-Gaussian (ShG), and cosine-Gaussian (ChG) profiles—in strongly nonlocal nonlinear media. The novelty lies in deriving exact dynamical solutions and stability criteria for each profile, enabling direct comparison of confinement efficiency, critical power, and robustness. Quantitative results show that ChG beams achieve the lowest critical power for self-trapping, with \(P_{\textrm{cr}}^{\textrm{ChG}}\) up to \(37\%\) lower than ShG and \(22\%\) lower than ThG at \(m=3\) , and maintain stable propagation up to 2.1 times their critical power threshold. ChG profiles also exhibit the broadest stability domain in the \((m, P_0)\) parameter space and the highest confinement efficiency, with efficiency values exceeding those of ThG and ShG by factors of 1.5 and 2.3, respectively, for \(\sigma =5\) . Spectral analysis reveals that increasing the profile order m leads to a 1.8 times broadening of the spectral energy for ChG beams compared to ThG. These findings establish ChG beams as optimal for robust, power-efficient self-trapped beam propagation in nonlocal nonlinear optical systems and provide universal design principles for advanced soliton-based photonic devices. The findings offer direct applicability in advanced soliton-based photonic devices including all-optical soliton routers, reconfigurable beam steering platforms in nematic liquid crystals, and hollow Gaussian pulse control in photonic crystal fibers. The ability to engineer beam symmetry and stability under nonlocality provides a foundation for robust nonlinear signal processing.
