  We consider the second Painlev\'e equation $$ u"(x)=2u^3(x)+xu(x)-\alpha, $$
where $\alpha $ is a nonzero constant. Using the Deift-Zhou nonlinear steepest
descent method for Riemann-Hilbert problems, we rigorously prove the
asymptotics as $x \to \pm \infty$ for both the real and purely imaginary
Ablowitz-Segur solutions, as well as the corresponding connection formulas. We
also show that the real Ablowitz-Segur solutions have no real poles when
$\alpha \in (-1/2, 1/2)$.
