On the large-time asymptotics of the defocusing mKdV equation with step-like initial data

It is concerned with the large-time asymptotics of the Cauchy problem of the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data subject to compact perturbations, that is, \begin{align*} q_{0}(x)-q_{0c}(x)=0, \ \text{for} \ |x|>N \end{align*} with some positive \(N\), where \begin{align*} q_{0c}(x)=\left\{ \begin{aligned} &c_{l}, \quad x\leqslant 0, &c_{r}, \quad x>0, \end{aligned} \right. \end{align*} and \(c_l>c_{r}>0\). It follows from the standard direct and inverse scattering theory that an RH characterization for the step-like problem is constructed. By performing the nonlinear steepest descent analysis, we mainly derive the large-time asymptotics in the each of four asymptotic zones in the \((x,t)\)-half plane.