  Exclusive diffractive processes, such as exclusive vector meson production,
serve as excellent probes of hadron structure within the perturbative regime of
quantum chromodynamics (QCD). The exclusive process involving light and heavy
vector mesons, $e p \rightarrow e V(V=J/\Psi,\rho,\phi)$, has been investigated
at the HERA accelerator facility. In this study, we focus on the theoretical
prediction of exclusive $J/\Psi$ and $\rho^0$ vector meson production.
Employing the color dipole description of deep inelastic scattering (DIS), we
calculate both the differential cross-section and total cross-section of
$J/\Psi$ and $\rho^0$ vector mesons, employing an analytical solution of the
Balitsky-Kovchegov (BK) equation. Furthermore, we present the ratio of the
longitudinal to the transverse cross-section for $J/\Psi$ and $\rho^0$ as a
function of $Q^2$. Two well-known vector meson wave function models, Boosted
Gaussian (BG) and Gaus-LC, have been integrated into our analysis, showing
slight sensitivity to the chosen vector meson wave functions. Our theoretical
predictions agree well with the available experimental data for vector meson
production. The analytical solution of the BK equation proves reliable for the
theoretical prediction of exclusive vector mesons within a specific range of
$Q^2$.
