The Painlevé-type asymptotics of defocusing complex mKdV equation with finite density initial data

We consider the Cauchy problem for the defocusing complex mKdV equation with finite density initial data \begin{align*} &q_t+\frac{1}{2}q_{xxx}-3|q|^2q_{x}=0,\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*} which can be formulated into a Riemann-Hilbert (RH) problem. With $\bar{\mathrm{\partial }}$-generation of the nonlinear steepest descent approach and a double scaling limit technique, in the transition region $$\mathcal{D}:=\{(x,t)\in \mathbb{R}\times {\mathbb{R}}^{+}\mid -C<(x\mathrm{/}(2t)+3\mathrm{/}2)t^{2\mathrm{/}3}<0,C\in {\mathbb{R}}^{+}\},$$ we find that the long-time asymptotics of the solution $q(x,t)$ to the Cauchy problem is associated with the Painlevé-II transcendents.