Long-time asymptotics for the defocusing Hirota equation with finite density initial data

Using the Riemann–Hilbert (RH) formulation of the Cauchy problem together with the ∂¯-generalized nonlinear steepest descent method, we derive the long-time asymptotics of the defocusing Hirota equation in the oscillatory regime. For ξ<−8, four phase points lie on the jump contour R, yielding the asymptotic expansion q(x,t)=−1+t−1/2h+O(t−3/4). Here, the leading term −1 represents the non-zero background; the t−1/2h term originates from the continuous spectrum; and the O(t−3/4) term accounts for the contribution of the pure ∂¯-RH problem.