Soliton Resolution and Asymptotic Stability of $N$-soliton Solutions for the Defocusing Kundu-Eckhaus Equation with Nonzero Boundary Conditions

In this paper, we address interesting soliton resolution, asymptotic stability of $N$-soliton solutions and the Painlev'e asymptotics for the Kundu-Eckhaus(KE) equation with nonzero boundary conditions. The key for proving these results is to establish the formulation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem and find its connection with RH problem of the defocusing NLS equation. With the $\bar\partial$-steepest descent method and the results of the defocusing NLS equation, we find complete leading order approximation formulas for the defocusing KE equation on the whole $(x,t)$ half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region, Zakharov-Shabat asymptotics in a solitonless region and the Painlev'e asymptotics in two transition regions.