  Previous studies on the generalized XY model have concentrated on the
equilibrium phase diagram and the equilibrium nature of distinct phases under
varying parameter conditions. We direct our attention towards examining the
systems evolution towards equilibrium states across different parameter values,
specifically by varying the relative strengths of ferromagnetic and nematic
interactions. We study the kinetics of the system, using the temporal
annihilation of defects at varying temperatures and its impact on the
coarsening behavior of the system. For both pure polar and pure nematic
systems, we observe temperature-dependent decay of the exponent, leading to a
decelerated growth of domains within the system. At parameter values where both
ferromagnetic and nematic interactions are simultaneously present, we show a
phase diagram highlighting three low-temperature phases : polar, nematic, and
coexistence, alongside a high-temperature disordered phase. Our study provides
valuable insights into the complex interplay of interactions, offering a
comprehensive understanding of the systems behavior during its evolution
towards equilibrium.
