Painlevé-type asymptotics for the defocusing Hirota equation in transition region

We consider the long-time asymptotics for the defocusing Hirota equation with Schwartz Cauchy data in the transition region. On the basis of direct and inverse scattering transform of the Lax pair of Hirota equations, we first express the solution of the Cauchy problem in terms of the solution of a Riemann–Hilbert problem. Further, we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region |x/t−(α2/3β)|t2/3≤M, where M is a positive constant. Our result shows that the long-time asymptotics of the Hirota equation can be expressed in terms of the solution of the Painlevé II equation.