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In this work, we consider the Cauchy problem for the defocusing Hirotaequation with a nonzero background \begin{align} \begin{cases}iq_{t}+\alpha\left[q_{xx}-2\left(\left\vertq\right\vert^{2}-1\right)q\right]+i\beta\left(q_{xxx}-6\left\vertq\right\vert^{2}q_{x}\right)=0,\quad (x,t)\in \mathbb{R}\times(0,+\infty),\\q(x,0)=q_{0}(x),\qquad \underset{x\rightarrow\pm\infty 1}{\lim} q_{0}(x)=\pm 1,\qquad q_{0}\mp 1\in H^{4,4}(\mathbb{R}). \end{cases} \nonumber \end{align}  According to the Riemann-Hilbert problem representation of the Cauchy problemand the $\bar{\partial}$ generalization of the nonlinear steepest descentmethod, we find different long time asymptotics types for the defocusing Hirotaequation in oscillating region and transition region, respectively. For theoscillating region $\xi<-8$, four phase points appear on the jump contour$\mathbb{R}$, which arrives at an asymptotic expansion,given by \begin{align}q(x,t)=-1+t^{-1/2}h+O(t^{-3/4}).\nonumber \end{align} It consists of threeterms. The first term $-1$ is leading term representing a nonzero background,the second term $t^{-1/2}h$ originates from the continuous spectrum and thethird term $O(t^{-3/4})$ is the error term due to pure $\bar{\partial}$-RHproblem. For the transition region $\vert\xi+8\vert t^{2/3}<C$, three phasepoints raise on the jump contour $\mathbb{R}$. Painlev\'{e} asymptoticsexpansion is obtained \begin{align}q(x,t)=-1-(\frac{15}{4}t)^{-1/3}\varrho+O(t^{-1/2}),\nonumber \end{align} inwhich the leading term is a solution to the Painlev\'{e} II equation, the lastterm is a residual error being from pure $\bar{\partial}$-RH problem andparabolic cylinder model.

Long time and Painlev\'{e}-type asymptotics for the defocusing Hirota equation with finite density initial data