  We considered the phantom cosmology with a lagrangian $\displaystyle
L=\frac{1}{\eta}[1-\sqrt{1+\eta g^{\mu\nu}\phi_{, \mu}\phi_{, \nu}}]-u(\phi)$,
which is original from the nonlinear Born-Infeld type scalar field with the
lagrangian $\displaystyle L=\frac{1}{\eta}[1-\sqrt{1-\eta g^{\mu\nu}\phi_{,
\mu}\phi_{, \nu}}]-u(\phi)$. This cosmological model can explain the
accelerated expansion of the universe with the equation of state parameter
$w\leq-1$. We get a sufficient condition for a arbitrary potential to admit a
late time attractor solution: the value of potential $u(X_c)$ at the critical
point $(X_c,0)$ should be maximum and large than zero. We study a specific
potential with the form of
$u(\phi)=V_0(1+\frac{\phi}{\phi_0})e^{(-\frac{\phi}{\phi_0})}$ via phase plane
analysis and compute the cosmological evolution by numerical analysis in
detail. The result shows that the phantom field survive till today (to account
for the observed late time accelerated expansion) without interfering with the
nucleosynthesis of the standard model(the density parameter
$\Omega_{\phi}\simeq10^{-12}$ at the equipartition epoch), and also avoid the
future collapse of the universe.
