Soliton resolution and asymptotic stability of N-solutions for the defocusing Kundu–Eckhaus equation with nonzero boundary conditions

Abstract

In this paper, we address interesting soliton resolution, asymptotic stability of N-soliton solutions and the Painlevé asymptotics for the Kundu-Eckhaus (KE) equation with nonzero boundary conditions iqt+qxx−2(∣q∣2−1)q+4β2(∣q∣4−1)q+4iβ∣q∣2xq=0,q(x,0)=q0(x)∼±1,x→±∞. 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 ∂¯ -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 Painlevé asymptotics in two transition regions.