Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions, which can be characterized using a Riemann-Hilbert (RH) problem through the inverse scattering transform. Using the \(\bar\partial\) generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem in the transition region \(|x/t+6|t^{2/3}< C\) with \(C>0\). The asymptotics is expressed in terms of the solution of the second Painlev\'{e} transcendent.