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

A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems