  The fusion mechanism of reactions involving even-even $^{112-124}$Sn, doubly
magic $^{132}$Sn, $^{208}$Pb as targets, and $^{64}$Ni as the projectile is
explored within the relativistic mean field (RMF) formalism. The main aim of
choosing these nuclei is to explore the correlation between the nuclear
incompressibility and the fusion cross-section. The nucleus-nucleus interaction
potential is calculated by folding the axially deformed nuclear densities and
the relativistic R3Y nucleon-nucleon (NN) potential obtained for the nonlinear
sets of NL3$^*$, hybrid, and NL1, which yield different values for various
characteristics of nuclear matter at saturation. The fusion barrier
characteristics obtained for different RMF parameterizations are further used
to calculate the cross-section within the $\ell$-summed Wong model. We found a
decrease in the barrier height and consequently, an increase in the
cross-section with a decrease in the incompressibility for all sets of
parameters considered. The calculated cross section is satisfactorily
consistent with the available experimental data for $^{64}$Ni+$^{208}$Pb
system. In contrast, the nuclear potentials obtained for NL3$^*$ and the hybrid
parameter sets underestimate the cross-section at below-barrier energies for
$^{64}$Ni+$^{112-124,132}$Sn reactions. This discrepancy between the
experimental data and the theoretical results for $^{64}$Ni+$^{112-124,132}$Sn
reactions can be correlated with the soft behaviour of the Sn isotopes. The
compressible nature of Sn-isotopes is inferred to lower the barrier height,
which further leads to enhancement of the experimental fusion and/or capture
cross-section at below-barrier energies. Thus, the NL1 parameter set with a
comparatively soft equation of state (EoS) is observed to be a better choice to
describe the sub-barrier nuclear fusion dynamics of reactions involving the
Sn-isotopes.
