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

We consider the Cauchy problem for the classical Hirota equation on the line with decaying initial data. Based on the spectral analysis of the Lax pair of the Hirota equation, we first expressed 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 $|\frac{x}{t} - \frac{α^2}{3β}|t^{2/3} \leq M$, $M$ 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 Painlevé $\mathrm{II}$ equation. Keywords: Hirota equation, steepest descent method, Painlevé $\mathrm{II}$ equation, long-time asymptotics.