Dynamics of $Z_N$ domain walls with bias directions

The spontaneous breaking of a discrete symmetry can lead to the formation of domain walls in the early Universe. In this work, we explore the impact of bias directions on the dynamics of $Z_N$ domain walls, mainly focusing on the $N=3$ model with a biased potential. Utilizing the Press-Ryden-Spergel method, we numerically investigate the dynamics of domain walls with lattice simulations. We find notable differences in the dynamics of domain walls due to bias directions. Our results indicate that the annihilation time depends not only on the vacuum energy difference $\delta V$ but also on bias directions described by the relative potential difference $\zeta$.