We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \(\begin{aligned}&q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \\&q(x,0)=q_{0}(x)\rightarrow \pm 1, \ \ x\rightarrow \pm \infty , \end{aligned}\) which can be characterized using a Riemann–Hilbert 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 for the defocusing mKdV equation in the transition region \(|x/t+6|t^{2/3}< C\) with \(C>0\) . The asymptotics can be expressed in terms of the solution of the second Painlevé transcendent.
