  We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in
the AKNS hierarchy of integrable equations. Our analysis is based on the
$\dbar$-steepest descent method. We consider RHPs arising from the inverse
scattering transform of the AKNS hierarchy with $H^{1,1}(\R)$ initial data. The
analysis will be divided into three regions: fast decay region, oscillating
region and self-similarity region (the Painlev\'e region). The resulting
formulas can be directly applied to study the long-time asymptotic of the
solutions of integrable equations such as NLS, mKdV and their higher-order
generalizations.
