Late-time phantom universe in ${\bf SO(1,1)}$ dark energy model with exponential potential

We discuss the late-time property of universe and phantom field in the SO(1,1) dark energy model for the potential $V=V_0e^{-\beta\Phi^\alpha}$ with $\alpha$ and $\beta$ two positive constants. We assume in advance some conditions satisfied by the late-time field to simplify equations, which are confirmed to be correct from the eventual results. For $\alpha<2$, the filed falls exponentially off and the phantom equation of state rapidly approaches -1. When $\alpha=2$, the kinetic energy $\rho_k$ and the coupling energy $\rho_c$ become comparable but there is always $\rho_k<-\rho_c$ so that the phantom property of field proceeds to hold. The analysis on the perturbation to the late-time field $\Phi$ illustrates the square effective mass of the perturbation field is always positive and thus the phantom is stable. The universe considered currently may evade the future sudden singularity and will evolve to de Sitter expansion phase.