Abstract
We study the long-time asymptotic behavior of oscillatory Riemann-Hilbert problems (RHPs) arising in the mKdV hierarchy (reducing from the AKNS hierarchy). Our analysis is based on the idea of partial derivative-steepest descent. We consider RHPs generated from the inverse scattering transform of the AKNS hierarchy with weighted Sobolev initial data. The asymptotic formula for three regions of the spatial- and temporal-dependent variables is presented in details.