The phase-shift analysis of proton-proton scattering has been studied by various research groups using realistic potentials that comprise multiple internal interactions mediated by pions and mesons, with a large number of parameters. The goal of the research is to construct inverse potentials for various \(\ell\) -channels of proton-proton(pp) elastic scattering using the 3-parameter Morse function in combination with atomic Hulth \(\grave{e}\) n utilizing the phase function method and the variational optimization technique. The implementation of variational optimization begins by randomly assigning initial values to the parameters of the Morse model. Using the Morse + atomic Hulth \(\grave{e}\) n potential as input, the phase equations for various \(\ell\) -channels are numerically solved using the RK-5 method to obtain the simulated Scattering Phase Shifts (SPS). We choose the mean squared error between the simulated and expected SPS as the cost function. Variational optimization iteratively adjusts potential parameters and re-evaluates the cost function until it achieves convergence. The analysis revealed that all the obtained scattering phase shifts across various \(\ell\) channels converge to a mean-squared error of \(\le\) 0.3. The computed cross-sections matched the experimental ones to less than 1% for energies up to 25 MeV. The scattering parameters also closely match the experimental data. The inverse potentials constructed for various \(\ell\) -channels using Morse+atomic Hulth \(\grave{e}\) n are on par with the currently available high-precision realistic potentials.
