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We consider the Cauchy problem for the defocusing complex mKdV equation witha finite density initial data \begin{align*}&q_t+\frac{1}{2}q_{xxx}-\left(|q|^2q\right)_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm1, \ x\to \pm\infty, \end{align*}  which can be formulated into a Riemann-Hilbert(RH) problem. With a$\bar\partial$-generation of the nonlinear steepest descent approach and adouble scaling limit technique, in the transition region$$\mathcal{D}:=\left\{(x,t)\in\mathbb{R}\times\mathbb{R}^+\big|-C<\left(\frac{x}{2t}+\frac{3}{2}\right) t^{2/3}<0, C\in\mathbb{R}^+\right\},$$ wefind that the long-time asymptotics of the solution $q(x,t)$ to the Cauchyproblem is associated with the Painlev\'{e}-II transcendents.

The Painlev\'{e}-type asymptotics of defocusing complex mKdV equation with finite density initial data