Soliton resolution and asymptotic stability of N-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 Painleve asymptotics for the Kundu-Eckhaus (KE) equation with nonzero boundary conditions iqt+qxx-2( divide q divide 2-1)q+4 beta 2( divide q divide 4-1)q+4i beta divide q divide 2xq=0,q(x,0)=q0(x)similar to +/- 1,x ->+/-infinity.The key to 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 the RH problem of the defocusing NLS equation. With the partial differential over bar -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 Painleve asymptotics in two transition regions.