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 ϑ(λ_i) (i = 0, 1) would like to be imaginary, which results in the \(\delta _{{{\rm{\lambda}}_i}}^0\) contains an increasing \(t{{\pm \,Im\vartheta ({{\rm{\lambda}}_i})} \over 2}\), and then the asymptotic behavior for nonlocal Hirota equation becomes differently.