  Under investigation in this work is the coupled Sasa-Satsuma equation, which
can describe the propagations of two optical pulse envelopes in birefringent
fibers. The Riemann-Hilbert problem for the equation is formulated on the basis
of the corresponding $5\times5$ matrix spectral problem, which allows us to
present a suitable representation for the solution of the equation. Then the
Deift-Zhou steepest descent method is used to analyze the long time behavior of
the coupled Sasa-Satsuma equation.
