  We consider the initial value problem for a new two-component Sasa-Satsuma
equation associated with the fourth-order Lax pair with decaying initial data
on the line. By utilizing the spectral analysis, the solution of the new
two-component Sasa-Satsuma system is transformed into the solution of a
fourth-order matrix Riemann-Hilbert problem. Then the long-time asymptotics of
the solution is obtained by means of the nonlinear steepest descent method of
Deift and Zhou for oscillatory Riemann-Hilbert problems. We show that there are
three main regions in the half-plane, where the asymptotics has qualitatively
different forms: a left fast decaying sector, a central Painleve sector where
the asymptotics is described in terms of the solution of a new coupled Painleve
II equation which is related to a fourth-order matrix Riemann-Hilbert problem,
and a right slowly decaying oscillatory sector.
