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|2)xq=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.
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