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

Abstract

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 i q t + q xx − 2 ( ∣ q ∣ 2 − 1 ) q + 4 β 2 ( ∣ q ∣ 4 − 1 ) q + 4 i β ∣ q ∣ 2 x q = 0 , q ( x , 0 ) = q 0 ( 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.