Long-time asymptotics for the reverse space-time nonlocal Hirota equation with decaying initial value problem: Without solitons

In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the $\vartheta(\lambda_{i})(i=0, 1)$ would like to be imaginary, which results in the $\delta_{\lambda_{i}}^{0}$ contains an increasing $t^{\frac{\pm Im\vartheta(\lambda_{i})}{2}}$, and then the asymptotic behavior for nonlocal Hirota equation becomes differently.